[{"data":1,"prerenderedAt":19800},["ShallowReactive",2],{"post-\u002Fblog\u002F2026\u002F0002-brief-biography-of-von-neumann":3,"i-ci:tag":19794,"i-mingcute:time-line":19798},{"post":4,"nextPost":292,"prevPost":302},{"id":5,"title":6,"body":7,"description":278,"draft":289,"enableComment":290,"extension":291,"image":278,"important":289,"location":292,"meta":293,"navigation":290,"ogImage":292,"path":294,"seo":295,"stem":296,"summary":297,"tags":298,"time":300,"__hash__":301},"blog\u002Fblog\u002F2026\u002F0002-brief-biography-of-von-neumann.md","冯 · 诺依曼小传",{"type":8,"value":9,"toc":277},"minimark",[10,14,18,21,28,31,34,37,40,43,46,49,52,55,58,61,64,67,70,73,76,79,82,85,88,91,94,97,100,103,106,109,112,115,118,124,127,130,133,136,139,142,145,148,151,154,157,160,163,166,169,172,175,178,181,184,187,190,193,196,199,202,205,208,211,214,217,220,223,226,229,232,235,238,241,244,247,250,253,256,259,262,265,268,271,274],[11,12,13],"h2",{"id":13},"布达佩斯",[15,16,17],"p",{},"匈牙利，1903 年 12 月 28 日，布达佩斯的一户殷实人家添了第一个男孩。父亲马克斯·诺依曼是位银行家，母亲玛吉特·肯恩出身富商之家。家里人叫他「扬奇」——匈牙利名字扬诺什的昵称。他还有两个弟弟，米哈伊和米克洛什。",[15,19,20],{},"扬奇从小就不太寻常——别的小孩还在学数数，他已经能用古希腊语跟父亲互相打趣了。他后来懂六种以上语言：匈牙利语、德语、法语、英语、拉丁语、古希腊语，晚年还学一点西班牙语。乌拉姆回忆，他用英语写论文时，已经能写出比大多数美国数学家更地道的英文散文。他学新语言的路子，跟解数学题是同一个路数——有同事回忆，他拿到一本新语言的语法书，先翻动词变位表，自己把规则推演出来，再拿例句验证。",[15,22,23],{},[24,25],"img",{"alt":26,"src":27},"童年时的约翰 · 冯 · 诺依曼","https:\u002F\u002Fimage-assets.dreams.plus\u002F202609020248182.png",[15,29,30],{},"六岁那年，他在心里算两个八位数的除法，速度快得让在座的客人怀疑是提前排练的把戏。他母亲有一次在饭桌上一时兴起，随口考他：「你说，你要是盯着窗外看，脑子里能想什么？」扬奇想都没想地回答出一堆答案，逗得全家人哈哈大笑——这大概是他这辈子第一次在饭桌上「表演」，后来这种饭桌高谈阔论的习惯，他保持了一辈子。",[15,32,33],{},"一九一三年，皇帝弗朗茨·约瑟夫册封马克斯为贵族，授予世袭的「玛尔吉泰」头衔——大致相当于德语里的「冯」。父亲自己没把这个头衔当回事，倒是儿子后来在德文论文上署起了「冯·诺依曼」的名字，一路用到了美国。",[15,35,36],{},"十岁那年，家里给他请了数学家教。没多久，一位名叫费凯特的匈牙利数学家登门找他讨论高等数学——据说聊到后来，这位比他大十几岁的年轻人被眼前的少年感动得眼眶发热。费凯特后来成了他学术生涯的引路人之一，两人还合写了他最早的一篇论文。",[15,38,39],{},"也是在十来岁的年纪，他一头扎进家里一套四十八卷本的《世界历史》，几个月就啃完了，之后养成一个爱好：拿当下报纸上的新闻，跟历史上某场战役类比，一本正经地讨论两边的战术优劣，仿佛在分析一盘棋局。他的记性好得被家人和师长传成奇谈——据说他能背下一整本小说，能记住电话簿里连续好几页的号码，谁想考他，报个页码就行。多年以后，普林斯顿一位专攻拜占庭史的教授私下承认，聊起拜占庭历史的细节，三十多岁的诺依曼懂得比他这个专业教授还多——而这只是诺依曼的一个业余爱好。",[15,41,42],{},"不过，扬奇不是散养出来的神童。马克斯给儿子请的不是普通家教，而是布达佩斯大学的年轻数学家和科学家，让他们定期来家里上课；诺依曼的弟弟后来回忆，家里餐厅的椅子常常被拉到书房去，因为「上课的人太多了」。父亲自己没受过多少正规科学训练，但他知道什么值得花钱——他给儿子买的，是一整套经过设计的私人教育。",[15,44,45],{},"这样的家教安排，在那个年代的布达佩斯并不罕见。二十世纪初，布达佩斯的一批犹太中产家庭把数学和语言教育当成头等大事，舍得为下一代花钱；诺依曼、维格纳、西拉德、特勒、冯·卡门——这些后来跟原子弹、氢弹、航空这些词紧紧绑在一起的名字——都出自同一个城市、相近的阶层。这群人中间后来流传一个玩笑：火星上的侦察兵在布达佩斯落脚，跟当地人通婚，留下孩子，又回了火星，于是有了这批「不像地球人」的科学家。玩笑归玩笑，背景是实的：诺依曼和维格纳上的是同一所路德中学，教过他们数学的是同一位老师——拉茨·拉斯洛。诺依曼不是孤例，他只是这一批人里算得最快、路走得最宽的一个。",[15,47,48],{},"家里原本盘算让他子承父业，将来接银行的班，至少学门「正经手艺」。至于数学，父亲倒也不反对，只是觉得那门手艺养活不了人。",[11,50,51],{"id":51},"从化学到量子力学",[15,53,54],{},"于是，他一边挂名在苏黎世联邦理工学院读化学工程，一边偷偷跑去旁听数学课，每学期还要跑回布达佩斯参加考试。两条腿走路，两头都没耽误：二十岁上下，他先后拿到了苏黎世的化学工程文凭和布达佩斯大学的数学博士学位，论文题目是《集合论的公理化》，那年他二十二岁。头一篇正式发表的论文，其实早在中学时代就出来了——一九二二年，还不满十八岁的他和费凯特合写了一篇文章，讨论一类极小多项式零点的位置，把切比雪夫多项式的一个经典结果往前推了一步。",[15,56,57],{},"父亲的心愿和儿子自己的兴趣，算是用一种曲线救国的方式两全了。那套化学工程的训练他后来几乎没再用过，倒是养成了他一辈子的习惯：把物理问题先翻译成数学，再拿数学去对付它。",[15,59,60],{},"博士毕业那年秋天，他拿了一笔洛克菲勒奖学金去了哥廷根，在大卫·希尔伯特身边做研究。那时候量子力学刚成形不过几年，海森堡的矩阵力学、薛定谔的波动力学、玻恩的概率诠释，各家表述五花八门，物理学家正吵得不可开交。诺依曼在哥廷根那几年，做的一桩费力的事：把量子力学翻译成严格的数学语言。他用希尔伯特空间搭起统一框架，讨论观测量的谱分解，还头一回把「测量」写成一个数学过程——一九三二年出版的德文专著《量子力学的数学基础》，成了这个领域绕不过去的一本书。这套「给年轻学科搭骨架」的本事，往后他还会用很多次。",[15,62,63],{},"一九三〇年九月，在哥尼斯堡的一场学术会议上，一位叫库尔特·哥德尔的年轻逻辑学家报告了一个令人不安的结果：任何足够强的一致公理系统里，都有既不能证明也不能证伪的命题——这就是后来著名的「不完备定理」。台下大多数听众没太当回事，诺依曼却当场就抓住了证明的要害。散会之后他顺着想下去，琢磨出一个哥德尔当时还没说透的推论：这样的系统连「自身是一致的」这件事都无法证明。他写了一封信给哥德尔，信里把这个结论完整地推了一遍。等哥德尔的回信到了，他才得知，哥德尔本人也已经发现了这个结果，正在把它写进次年发表的文章里。诺依曼没有争这个先后，后来还在公开场合把功劳归给哥德尔。",[15,65,66],{},"在柏林和汉堡当讲师的那些年，他一边教课一边写，算子理论、遍历理论、测度论，到处都有他插手的痕迹。",[11,68,69],{"id":69},"研究院与博弈论",[15,71,72],{},"一九三〇年，他应普林斯顿大学的邀请跨过大西洋去讲学。一九三三年，新成立的普林斯顿高等研究院聘他做创始教授——第一批六位教授里，有爱因斯坦，有外尔，有刚从纳粹德国跑出来的哥德尔。诺依曼是最年轻的一个。",[15,74,75],{},"三十年代是他学术上最密集的一段时间。他跟弗朗西斯·默里合作，把「算子代数」这门学科从无到有地搭起来——后来这套东西被称作「冯·诺依曼代数」，至今仍是数学和量子物理交叉处的重要工具。他还把集合论里「序数」的定义做了现代化处理，证明了紧群上哈默测度的唯一性；一九三二年，他又证明了动力系统里一个关于时间平均的定理——后来被称作「平均遍历定理」，成了统计物理的基本工具。光是在普林斯顿的头十年，他横跨的领域多得让同行咋舌——但「咋舌」是我们这些后来的读者会用的词，当时他身边人的反应往往是：又来了。",[15,77,78],{},"一九二八年，他还在柏林时就发表过一篇讨论博弈的文章，研究两人零和博弈，证明了著名的「极小极大定理」——玩家在最坏情况下求最好结果的策略，跟对方的策略同时达成平衡。这篇论文当时在数学圈里没掀起多少波澜。真正让它出圈，要等他和一位经济学家奥卡·莫根施特恩的合作。莫根施特恩是奥地利人，一九三八年逃到普林斯顿，两人一拍即合，一九四四年出版了《博弈论与经济行为》——一本经济学史和数学史上都绕不过去的大书。把「博弈」变成一门有严格结构的学科，跟当年给量子力学搭骨架，是同一个路数。",[15,80,81],{},"研究院边上，他的生活渐渐固定下来。家里几乎每周都要开派对，客厅里高朋满座，他自己端着酒杯一圈一圈地敬酒讲笑话，笑话大多带点荤腥，气氛总能被他搅得热闹非凡。据好友乌拉姆回忆，他喜欢昂贵的西装、烈酒和跑车，是那种谁都愿意请去撑场面的座上宾。",[15,83,84],{},"他跟哥德尔的交情值得一提。哥德尔是出了名的孤僻，在研究院里少有人能跟他真正说上话，诺依曼是少数几个。两人常在普林斯顿的林间小道上一起散步，一聊就是很久——聊的什么，旁人很少知道，大约是逻辑与数学的底细。研究院里的气氛并不总是一团和气：他跟外尔能合作，跟爱因斯坦之间却谈不上热络，两人对许多事情的看法相去甚远，尤其是在战争与核武器的问题上。",[15,86,87],{},"还有一件后来才被翻出来的小事。一九三八年春天，一位名叫艾伦·图灵的英国青年在普林斯顿完成了他的博士论文——那篇论文后来被称为「图灵机」的构想。诺依曼读过之后，提出让他留在高等研究院给自己当助手，年薪一千五百美元。图灵谢绝了，他说剑桥国王学院的职位更合意，然后乘船回了英国。多年以后人们回头翻这段旧账，常常感叹：如果当时图灵留了下来，计算机的早期历史也许会换一副样子。",[15,89,90],{},"一九三七年，他入了美国籍。也是那段时间，他离了婚，娶了第二任妻子克拉拉·丹。第一段婚姻的另一位主角玛丽埃特是布达佩斯人，两人婚后有个女儿玛丽娜，生于一九三五年。离婚后，玛丽娜先在母亲身边长大，后来又回到父亲身边。分隔两地的时候，他给女儿写很长的信；玛丽娜在回忆录里写道，有一封信，头八页都在劝她别急着结婚，后面才转到她的论文，再扯到他自己对历史的看法，顺手引了几段法文史料。玛丽娜后来读了经济学，一九七二年被尼克松任命为总统经济顾问委员会成员——是头一个坐上那个位子的女性。她晚年给回忆录起名《火星人的女儿》，写的就是前面那批「火星人」里的她父亲。",[15,92,93],{},"战争还没打起来，他的顾问身份就先忙上了。一九三七年之后，他先后给陆军和海军当顾问，处理弹道、炸药和空气动力学的问题。他参与改进了「聚能装药」——一种把炸药能量聚焦成射流来击穿装甲的设计，后来成了反坦克武器的原理。这些事他做得不声不响，连研究院里的同事都未必清楚。同一时期，他那几个布达佩斯老同乡也在为同一件事奔走——一九三九年，西拉德、维格纳和特勒三个人起草了一封给罗斯福的信，借爱因斯坦的名字签了出去，提醒美国政府注意铀裂变的前景；诺依曼没有在这封信上署名，但布达佩斯这群人之间，消息向来是通的。",[11,95,96],{"id":96},"原子弹的算盘",[15,98,99],{},"一九四三年，奥本海默把诺依曼请到了洛斯阿拉莫斯。名义上他是顾问，实际上他很快成了原子弹设计里「算」这一环的关键人物。",[15,101,102],{},"原子弹的临界质量怎么算？炸药怎么聚爆才对称？冲击波在水里、在空气里怎么传播？这些在诺依曼到来之前，许多还停留在估算阶段。他把流体力学和数值方法带进了洛斯阿拉莫斯，一起带进去的还有他那张出了名的「活计算器」。聚爆方案最初是另一位物理学家奈德迈耶提出的大胆设想，但把设想变成能用的设计，需要算清楚球面聚爆的流体力学细节——诺依曼接过这一摊，跟化学家基斯塔科夫斯基合作，把炸药透镜的配方和几何一点点调出来。他还算过另一种打法：把核弹在高空引爆，让冲击波在更大的面积上均匀压下来，造成的破坏比地面引爆大得多。广岛和长崎的投弹高度，就参考了这套计算。",[15,104,105],{},"一九四五年七月十六日凌晨，新墨西哥州阿拉莫戈多沙漠，人类第一次核爆试验——「三位一体」——在那里进行。诺依曼就站在观看的人群里，看着那颗以他的方程为基础的炸弹把自己烧成白光。他回去之后说了什么，没有可靠的记载；能确认的是，他给这次试验做过大量的计算，也为此吸收了不少辐射。在洛斯阿拉莫斯那段日子，他跟费米、费曼这些人在食堂里一桌吃饭、插科打诨，照样是那身讲究的打扮。原子弹的严肃和这些日常混在一起，在他身上并不显得矛盾。",[15,107,108],{},"战后他留在这个圈子里。氢弹的「超级」方案，他是最早的推动者之一；一九四六年他跟乌拉姆一起，在洛斯阿拉莫斯继续做数值计算。这些工作后来凝结成了他冷战幕僚生涯的一部分。",[11,110,111],{"id":111},"从算数到思考",[15,113,114],{},"战争还没结束，另一条线索已经悄悄接上了。",[15,116,117],{},"一九四四年夏天，在阿伯丁的火车站台上，诺依曼碰上了陆军派到宾夕法尼亚大学的联络官赫曼·哥德斯坦。哥德斯坦随口说起，大学里正在建造一台叫 ENIAC 的电子计算机——一万八千个真空管，用来算火炮射表。诺依曼听完，没过多久就出现在摩尔学院的每次设计会上。那一年正是曼哈顿计划最紧的时候，他两头跑；他很快意识到，这种「机器自动算」的路子，正是洛斯阿拉莫斯需要的。",[15,119,120],{},[24,121],{"alt":122,"src":123},"埃尼阿克计算机","https:\u002F\u002Fimage-assets.dreams.plus\u002F202609020249186.png",[15,125,126],{},"一九四五年六月，他起草了一份报告，题目叫《EDVAC 报告初稿》，把「存储程序」的思想写成白纸黑字——程序和数据一样放进内存，机器读一条执行一条，不必再靠插拔接线来「改程序」。",[15,128,129],{},"这份打印稿在圈子里私下流传，日后被称作「冯·诺依曼架构」的雏形。",[15,131,132],{},"严格说，「存储程序」是小组讨论的产物，不全是他的发明；但把它提炼成一份清晰的文件，又一路推动它付诸实现的，是他。这一步的关键不在算得快，而在想得远：机器从此不只是算数的工具，程序本身也成了可以被读写的东西。",[15,134,135],{},"战后他在高等研究院主持电子计算机项目，一九五二年，那台「IAS 机」正式跑了起来。这台机器的图纸流传出去，成了好几家机构机器的蓝本——兰德公司照着造的那台，直接被人叫作「约翰尼艾克」。",[15,137,138],{},"为了让机器有正经事干，他和乌拉姆、梅特罗波利斯一起，把一种用随机数抽样逼近复杂计算的「蒙特卡洛方法」搬上了机器，用来算中子的扩散。",[15,140,141],{},"他一边算，一边开玩笑：任何想用确定性办法产生随机数的人，都是「活在罪孽之中」。这句玩笑后来成了统计学家们的名言。顺带一提，他的第二任妻子克拉拉·丹也是这台机器背后的早期程序员之一，替蒙特卡洛计算写过代码。",[15,143,144],{},"他还惦记着用机器预报天气。四十年代他就反复对人说，天气预报是计算机最有价值的用途之一，甚至把人工造雨、控制气候也算进计算的版图。一九五二年前后，他拉了几位气象学家，真的在 IAS 机上跑起了数值天气预报——那台机器算一次要用很久，但方向算是立住了。",[15,146,147],{},"从「机器能算什么」到「机器到底是什么」，这条路他一路走到了底。他琢磨「自动机」能不能自我复制，设计过一个理论上能复制自身的自动机，还在一次研讨会上做过一场题为《自动机的逻辑理论》的演讲，讨论机器与人脑的关系。",[15,149,150],{},"一九五二年一月，他在加州理工连讲五讲，题目是《概率逻辑与由不可靠元件合成可靠有机体》。灵感来自脑子：神经元单个来看并不牢靠，可整颗脑子照样运转稳定——问题是，怎么用一堆不可靠的零件，拼出一台可靠到能算能记的机器。",[15,152,153],{},"这份讲稿后来在《自动机研究》上刊出，成了后世「容错计算」最早的理论基础之一。他一辈子都在跟「系统内部的不可靠」打交道，从集合论里的不完备，到神经元的不稳定，说到底像是同一件事。",[15,155,156],{},"这些想法他生前没有写完，身后被整理成《自复制自动机理论》一书。他给耶鲁预备的最后一份讲稿，题目就叫《计算机与人脑》，讲的还是同一个执念：机器和脑子，到底谁跟谁学。",[15,158,159],{},"有一次，记者问他正在想什么。他说：「我在想一件比炸弹重要得多的事——我在想计算机。」这话流传得很广，因为说这话的人，炸弹和计算机两样都参与得很深。",[11,161,162],{"id":162},"冷战幕僚",[15,164,165],{},"同一时期，他的另一副面孔也在忙碌。",[15,167,168],{},"学术这边，他把博弈论从桌上的游戏推向对人类决策的一般描述。跟莫根施特恩合著的《博弈论与经济行为》一九四四年出版，一开始读者寥寥——经济学家嫌它太数学，数学家嫌它太经济学。",[15,170,171],{},"诺依曼没太在意，照样在各种场合宣传这套东西「能描述一切有利益冲突的人类活动」。一九四七年十月，一个叫乔治·丹齐格的年轻人带着一道卡了自己很久的线性规划题来见他。",[15,173,174],{},"诺依曼看了一眼，说了句「哦，这个啊」，然后当场在黑板前面讲了一个多小时，把线性规划的「对偶性」从头推到尾——这个结论，丹齐格之前翻遍了文献也没找到。丹齐格后来回忆，自己当场就学会了法卡斯引理和对偶理论。而这场即兴讲座的另一个副产品，是把博弈论和线性规划这两种此前不搭界的数学连到了一起。",[15,176,177],{},"公共事务这边，他越来越深地卷进冷战。他是空军科学顾问委员会的成员，是兰德公司的顾问——兰德公司在加州的圣莫尼卡，他战后常去那边——还替原子能委员会出谋划策。",[15,179,180],{},"博弈论在这里找到了最严肃的用武之地：核威慑，本质上被他说成一场玩家之间不停盘算对方底牌的博弈。他的政治立场之强硬，在当时的科学家里也属于突出的那一档——一九五〇年，他私下对不少人讲过，美国应该趁苏联还没造出氢弹，先发制人。",[15,182,183],{},"这个主张在研究院的同僚里并不讨喜，研究院里也有不少人跟他观点相左，但在白宫和五角大楼的走廊里，他的话很受用。",[15,185,186],{},"一九五四年，奥本海默的安全听证会开庭。诺依曼和奥本海默在氢弹问题上站得正相反，可他还是去做了证人。听证官问他对奥本海默的忠诚怎么看，他说：毫无疑问，没有任何怀疑。他又当着听证官的面把两件事分开——反对氢弹是一回事，忠诚是另一回事；他还提醒，这套安全体系有个危险，就是把技术上的不同意见，误当成政治上的异心。听证会的结果没有因为他的证词而改变：奥本海默还是被吊销了安全许可，从此离开了决策圈。",[15,188,189],{},"同年，他主持的一个委员会（因为代号而被称为「茶壶委员会」）向空军提交了一份报告，建议美国大力发展洲际弹道导弹——后来美国的第一代洲际导弹，多少都算这份报告的产物。一九五五年，艾森豪威尔把他任命为原子能委员会委员——委员会里像他这样以科学家身份直接坐进决策层的，并不多见。",[15,191,192],{},"他经手的事务，大多与核武器的设计和导弹的部署有关：他那个委员会算过一笔账，一枚能装进洲际导弹的一兆吨氢弹，足以把两英里内的一切抹平；于是氢弹的小型化跟洲际导弹一起，成了他这边推进的两件事。",[15,194,195],{},"一九五六年，比基尼环礁上进行了新的一轮核试验，第一枚「袖珍氢弹」就是在那轮试验里炸响的——那时候，他坐在委员会里，离权力中心很近。据说不管对面坐着的是将军还是总统，他都能一样谈笑风生、自信从容；艾森豪威尔后来回忆过，给他颁发总统自由勋章那次，这位科学家的风趣谈吐给自己留下了很深的印象。",[11,197,198],{"id":198},"灵感",[15,200,201],{},"工作之外，穿着讲究到有点固执：常年穿着马甲，口袋里插一块手绢，外套的扣子永远扣得整整齐齐——即便是在新墨西哥沙漠里搞原子弹的洛斯阿拉莫斯，别人穿着短袖短裤在酷热里晃悠，他也照样西装革履。他喜欢精巧的小玩意，复杂的机械玩具、稀奇古怪的小装置，只要摆在他面前，他都要摆弄一番。",[15,203,204],{},"工作起来他是不折不扣的工作狂，一个问题不解决，他能一直死磕下去，半夜里灵感一来，翻身就爬起来接着算。可一旦离开书桌，他对生活琐事的粗心大意也是出了名的。据说有一次，他生病卧床的妻子想让他倒杯水，诺依曼却愣是想不起杯子放在哪儿——而那时候他已经在那栋房子里住了整整十七年。",[15,206,207],{},"开车对他来说更是一场持续多年的灾难。他几乎每年都要报废一辆车，熟人们甚至专门给他反复出事故的那个路口起了个外号，叫「冯·诺依曼转角」。他喜欢把车开得飞快，据他身边人的说法，那是把拥挤的路段当成一道多体问题，边开边在心里算最优路线——至于算得好不好，看事故率就知道。",[15,209,210],{},"他其实从没考下过驾照，连着几次没过之后，还是在妻子的建议下使了点手段，才拿到执照。他开的是凯迪拉克，问他为什么挑这个牌子，他说：因为没人肯卖给我坦克。据说他跟保险公司解释事故经过时说过一句流传很广的话：路边的树本来排得整整齐齐地朝我迎面而来，一直都挺好的，直到有一棵没躲开。",[15,212,213],{},"他食量也惊人。在洛斯阿拉莫斯搞原子弹那几年，他常常开车一百二十英里，专程跑去一家墨西哥餐馆撮一顿，吃完再开一百二十英里回来——对他而言，这大概不算什么特别麻烦的事。",[15,215,216],{},"他的算术快到经常让身边人下不来台。他上课喜欢直接在黑板上写答案，中间的推导过程一概省略，学生们常常一脸茫然。有一回一个学生不死心，追问他这道题是不是还有别的解法，诺依曼愣了一下，一本正经地点头说：「有。」然后就没了下文——那个所谓「另一种解法」，大概永远留在他自己脑子里了。",[15,218,219],{},"还有一次，麻省理工的一个学生在走廊里拦住他，请教一道积分题。诺依曼瞄了一眼题目，脱口而出答案。学生说，教授，我知道书后面写的答案就是这个，我是想请您讲讲怎么算出来的。诺依曼又看了一眼题目，想了想，把答案又报了一遍。学生急了，反复解释自己要的是过程。诺依曼摊摊手，一脸无辜地说：「你还想怎样，我可是用了两种不同的方法才验算出这个答案的。」",[15,221,222],{},"他的心算速度还成就了一个流传很广的派对段子：一次晚宴上，女主人出了道题考他——两列火车相距一英里，各以每小时三十英里的速度迎面开来，一只苍蝇栖在其中一列车头，以每小时六十英里的速度朝对面飞去，碰到之后立刻折返，如此来回，直到两车相撞，问苍蝇一共飞了多少英里。",[15,224,225],{},"大多数懂点数学的人，第一反应是去算一个无穷级数求和，麻烦但也不难；诺依曼几乎没停顿就报出了正确答案。提问的人有点失望地说，其实有个巧办法——先算出两车相撞要多久（正好一分钟），再乘以苍蝇的速度就行，用不着算级数。诺依曼有点意外地反问：「可是我刚才算的就是级数啊。」",[15,227,228],{},"还有个流传更广的故事：福特汽车厂里一台发电机坏了，工程师们查了半天也没找出毛病，慕名请诺依曼来看。他围着机器转了一圈，用粉笔在外壳上画了一条线，让工人从那儿拆开检查——问题果然就出在那儿。福特问他要多少钱，诺依曼开了一张五千美元的账单，还附上明细：「画一条线，一美元；知道该画在哪儿，四千九百九十九美元。」（这则轶事在另一位工程师身上也有相似的版本，流传得久了，人们总爱把它安在算得最快的人头上。）",[15,230,231],{},"还有一件小事。象棋这件事，他其实也早就想过了。他一九二八年的极小极大定理，写的是两个人如何在博弈里各取最优；后来计算机下棋所用的那套搜索方法，追根溯源，都是从那棵博弈树长出来的。一九五六年，洛斯阿拉莫斯的一台机器第一次用它下赢了业余棋手。他连这个都已经想过了。",[15,233,234],{},"战后，一位在洛斯阿拉莫斯工作的物理学家碰到一道棘手的题，跑去请教这位顾问。诺依曼说：简单，用特征线法就能解。物理学家老老实实说，我不懂特征线法。诺依曼看着他：「年轻人，在数学里，你不需要理解什么，你只需要习惯它们。」这句话后来被记了下来，成了许多数学系学生自我安慰的箴言。",[11,236,237],{"id":237},"病床上的最后一年",[15,239,240],{},"一九五五年，诺依曼被查出癌症——多半是骨癌，具体是什么，各种说法都有，但大家普遍认为，跟他这些年没少亲临核试验现场、接触辐射脱不了关系。",[15,242,243],{},"诊断没有让他停下来。他坐着轮椅照常去五角大楼开会，担任的原子能委员会委员一直当到生命的最后，躺在病床上还在口述那部没写完的手稿。",[15,245,246],{},"病床边的一些细节，是身边人后来才慢慢讲出来的。一向对宗教毫不在意的他，主动请来一位天主教神父，安排了临终仪式——不过那位神父私下说，自己也说不准这场仪式究竟有没有真的抚慰到这位科学家。",[15,248,249],{},"这件事让很多老朋友意外——他这辈子对宗教既不信，也不讨论，更不反对，只是像对待一套不感兴趣的数学公理一样，敬而远之。",[15,251,252],{},"诺依曼对仪式的配合，更像是完成一项程序：到了这个阶段，该走的步骤走一遍。他问过诺依曼是否想谈谈上帝。诺依曼的回答很短，大意是：如果有一种更高级的存在，它大概不会用我们这种语言说话。",[15,254,255],{},"这话既是拒绝，也没有完全关上。像他晚年的很多判断一样——保留可能性，不给出承诺。",[15,257,258],{},"乌拉姆后来回忆，诺依曼在病中有一阵子确实表现出对死亡的恐惧，那种恐惧比周围人预想的更具体，也更不「体面」。他会长时间地计算自己还剩多少时间，问医生每种治疗方案的概率，像在评估一个项目的失败风险。死亡对他而言不是哲学问题，是一道突然摆在面前的、信息不完备的决策题。",[15,260,261],{},"而宗教能提供的，恰恰是他一生都在绕开的那种东西：不确定、不可验证、不要求算清楚。他的理性是他最锋利的工具，也是他的边界。临终前他愿意走一遍仪式，却无法交出那个「信任的跳跃」。",[15,263,264],{},"这大概是一个一生都在把世界翻译成数学的人，最后发现，有些东西翻译不过去。这个人太理性了，一生理性的他，宗教抚慰很难走进他的内心。",[15,266,267],{},"因为他脑子里装着太多国防机密，弥留之际，军方甚至派了警卫守在病房门口，生怕他意识模糊时无意间说漏了嘴。",[15,269,270],{},"乌拉姆在他去世后写的纪念文章里，回忆了一次谈话：诺依曼谈到技术的加速进步，说人类的处境正在逼近某种「本质上的奇点」——越过那个点之后，按我们熟悉的方式运转的人类事务，恐怕难以继续下去。这话经由乌拉姆的笔传下来，成了「技术奇点」这个概念最早的出处之一。",[15,272,273],{},"一九五七年二月八日，诺依曼在华盛顿的沃尔特·里德陆军医院去世，享年五十三岁，葬在新泽西州普林斯顿。他女儿玛丽娜后来在回忆录里写道，父亲弥留之际说过一句话：「没有人真正理解另一个人。」——这是那种让人没法反驳、也让人没法安心的句子。",[15,275,276],{},"去世六年后，同样来自匈牙利的诺贝尔物理学奖得主尤金·维格纳在一次演讲后被人问：匈牙利怎么在同一个年代冒出这么多天才？维格纳想都没想地回答——只有冯·诺依曼一个人是天才，别人不过是脑子比较好使的普通人罢了。据说在场没人反驳这句话，尽管诺依曼这辈子从没拿过什么重量级的科学奖章。",{"title":278,"searchDepth":279,"depth":279,"links":280},"",2,[281,282,283,284,285,286,287,288],{"id":13,"depth":279,"text":13},{"id":51,"depth":279,"text":51},{"id":69,"depth":279,"text":69},{"id":96,"depth":279,"text":96},{"id":111,"depth":279,"text":111},{"id":162,"depth":279,"text":162},{"id":198,"depth":279,"text":198},{"id":237,"depth":279,"text":237},false,true,"md",null,{},"\u002Fblog\u002F2026\u002F0002-brief-biography-of-von-neumann",{"title":6,"description":278},"blog\u002F2026\u002F0002-brief-biography-of-von-neumann","从 1903 年布达佩斯的银行家之子，到 1957 年华盛顿病床上的原子能委员。他留下的不是一尊天才的神像，而是一道冷峻的背影——一个快得让时代跟不上的头脑，一个把理性用到极致、也走到理性尽头的人。",[299],"thoughts","2026-09-02T02:45:47+08:00","KFRWCJxndsMPsD1egcxGn8kEmxXpjHfYKLPgx7E-qNw",{"id":303,"title":304,"body":305,"description":309,"draft":289,"enableComment":290,"extension":291,"image":292,"important":289,"location":19784,"meta":19785,"navigation":290,"ogImage":292,"path":19786,"seo":19787,"stem":19788,"summary":19789,"tags":19790,"time":19792,"__hash__":19793},"blog\u002Fblog\u002F2026\u002F2026-08-22-are-local-minima-rare-in-high-dimensions.md","高维非凸优化的「鞍点主导」",{"type":8,"value":306,"toc":19765},[307,310,313,331,334,337,340,424,427,431,763,1695,1982,3348,3471,4136,4210,4707,5123,5364,5367,5783,5819,7138,7199,7480,7483,7486,7489,8063,8840,8906,8911,8914,8917,8920,9061,9223,9311,9595,9663,9962,10331,10334,10681,10683,11182,11185,11402,11463,11474,11477,11480,11483,11487,11525,11837,11869,11873,12030,12153,12156,12284,12287,13171,13308,13441,13444,13516,13937,14124,14320,14381,14864,14870,14874,14879,15074,15407,15468,16128,16243,16297,16610,16709,16784,16816,16819,16824,16828,16831,16966,17155,17316,17322,17609,17829,17832,17836,17843,18640,18643,18869,18872,19098,19211,19214,19217,19586,19593,19596,19599,19602,19613,19616,19619,19679,19740,19743,19748,19751,19755],[15,308,309],{},"深度学习中，Roon 说过一句话，被很多人当作信条：最大的 whitepill 是：局部极小值在高维空间中很罕见。也就是说，你的模型很难卡在局部最优里。但它到底有没有道理？如果有道理，道理又在哪里？",[15,311,312],{},"这句观点并非对每个光滑函数都字面成立。比如凸函数可能只有一个全局极小值，压根没有鞍点。它的精确版本需要一个「景观的概率模型」，以及退化与非退化临界点之间的区分。这篇文章会把这句直觉，分三步严格化：",[314,315,316,320,328],"ol",{},[317,318,319],"li",{},"对 Morse 函数而言，临界点是孤立的，因此在所处区域中占据零体积。",[317,321,322,323,327],{},"在非退化临界点的条件下，局部极小值要求 Hessian ",[324,325,326],"strong",{},"正定","，即所有 N 个独立曲率方向都为正。",[317,329,330],{},"在 Hessian 表现得像大型中心化随机对称矩阵的随机景观模型中，正定是指数级罕见的事件，鞍点则压倒性地常见。",[15,332,333],{},"Kac–Rice 公式为这一现象在高斯随机场中提供了严格的计数框架，Morse 理论则解释了为什么「指数」是临界点的自然拓扑不变量。下面我们一步步把这条链路走完。",[11,335,336],{"id":336},"高维的地形景观",[15,338,339],{},"我们习惯把优化问题想象成一幅二维或三维的地形图：有山峰（极大值）、山谷（极小值）、还有山口（鞍点，像马鞍一样一边上一边下）。这种直觉在低维时很管用。",[15,341,342,343,391,392,420,421],{},"但当维度 ",[344,345,349,371],"span",{"className":346,"translate":348},[347],"katex","no",[344,350,353],{"className":351},[352],"katex-mathml",[354,355,357],"math",{"xmlns":356},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[358,359,360,367],"semantics",{},[361,362,363],"mrow",{},[364,365,366],"mi",{},"N",[368,369,366],"annotation",{"encoding":370},"application\u002Fx-tex",[344,372,376],{"className":373,"ariaHidden":375},[374],"katex-html","true",[344,377,380,385],{"className":378},[379],"base",[344,381],{"className":382,"style":384},[383],"strut","height:0.6833em;",[344,386,366],{"className":387,"style":390},[388,389],"mord","mathnormal","margin-right:0.109em;"," 变得很大——几千、几百万，甚至更多——这幅图景就开始「失真」。想象你站在高维空间的一个点上：此时脚下不再只有「前后左右」两个方向，而是有 ",[344,393,395,408],{"className":394,"translate":348},[347],[344,396,398],{"className":397},[352],[354,399,400],{"xmlns":356},[358,401,402,406],{},[361,403,404],{},[364,405,366],{},[368,407,366],{"encoding":370},[344,409,411],{"className":410,"ariaHidden":375},[374],[344,412,414,417],{"className":413},[379],[344,415],{"className":416,"style":384},[383],[344,418,366],{"className":419,"style":390},[388,389]," 个相互独立的方向可以移动。要说这个点是一个真正的谷底（局部极小值），你必须保证：",[324,422,423],{},"无论往哪个方向迈一小步，海拔都会升高。",[15,425,426],{},"这个直觉在数学上如何精确化？我们从最基础的定义开始。",[11,428,430],{"id":429},"临界点梯度与-hessian","临界点、梯度与 Hessian",[15,432,433,434,563,564,702,703,762],{},"设 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是二阶连续可微函数，记作 ",[344,565,567,606],{"className":566,"translate":348},[347],[344,568,570],{"className":569},[352],[354,571,572],{"xmlns":356},[358,573,574,603],{},[361,575,576,578,581,590,594,600],{},[364,577,448],{},[450,579,580],{},"∈",[454,582,583,586],{},[364,584,585],{},"C",[587,588,589],"mn",{},"2",[450,591,593],{"stretchy":592},"false","(",[454,595,596,598],{},[364,597,459],{"mathvariant":458},[364,599,366],{},[450,601,602],{"stretchy":592},")",[368,604,605],{"encoding":370},"f \\in 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假设保证了 Hessian 良定义且对称。",[764,765,766,1044],"blockquote",{},[15,767,768,771,772,800,801,1043],{},[324,769,770],{},"定义 2.1（梯度）","。函数 ",[344,773,775,788],{"className":774,"translate":348},[347],[344,776,778],{"className":777},[352],[354,779,780],{"xmlns":356},[358,781,782,786],{},[361,783,784],{},[364,785,448],{},[368,787,448],{"encoding":370},[344,789,791],{"className":790,"ariaHidden":375},[374],[344,792,794,797],{"className":793},[379],[344,795],{"className":796,"style":479},[383],[344,798,448],{"className":799,"style":483},[388,389]," 在 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2.2（临界点）","。若 ",[344,1704,1706,1733],{"className":1705,"translate":348},[347],[344,1707,1709],{"className":1708},[352],[354,1710,1711],{"xmlns":356},[358,1712,1713,1730],{},[361,1714,1715,1722,1724],{},[822,1716,1717,1719],{},[364,1718,815],{},[364,1720,1721],{},"c",[450,1723,580],{},[454,1725,1726,1728],{},[364,1727,459],{"mathvariant":458},[364,1729,366],{},[368,1731,1732],{"encoding":370},"x_c \\in 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满足",[344,1830,1832],{"className":1831,"translate":348},[1047],[344,1833,1835,1867],{"className":1834,"translate":348},[347],[344,1836,1838],{"className":1837},[352],[354,1839,1840],{"xmlns":356,"display":1056},[358,1841,1842,1864],{},[361,1843,1844,1846,1848,1850,1856,1858,1860,1862],{},[364,1845,1064],{"mathvariant":1063},[364,1847,448],{},[450,1849,593],{"stretchy":592},[822,1851,1852,1854],{},[364,1853,815],{},[364,1855,1721],{},[450,1857,602],{"stretchy":592},[450,1859,818],{},[587,1861,1096],{},[450,1863,831],{"separator":375},[368,1865,1866],{"encoding":370},"\\nabla f(x_c) = 0,",[344,1868,1870,1937],{"className":1869,"ariaHidden":375},[374],[344,1871,1873,1876,1879,1882,1885,1925,1928,1931,1934],{"className":1872},[379],[344,1874],{"className":1875,"style":884},[383],[344,1877,1064],{"className":1878},[388],[344,1880,448],{"className":1881,"style":483},[388,389],[344,1883,593],{"className":1884},[668],[344,1886,1888,1891],{"className":1887},[388],[344,1889,815],{"className":1890},[388,389],[344,1892,1894],{"className":1893},[513],[344,1895,1897,1917],{"className":1896},[517,900],[344,1898,1900,1914],{"className":1899},[521],[344,1901,1903],{"className":1902,"style":1761},[525],[344,1904,1905,1908],{"style":910},[344,1906],{"className":1907,"style":533},[532],[344,1909,1911],{"className":1910},[537,538,539,540],[344,1912,1721],{"className":1913},[388,389,540],[344,1915,924],{"className":1916},[923],[344,1918,1920],{"className":1919},[521],[344,1921,1923],{"className":1922,"style":931},[525],[344,1924],{},[344,1926,602],{"className":1927},[701],[344,1929],{"className":1930,"style":488},[487],[344,1932,818],{"className":1933},[492],[344,1935],{"className":1936,"style":488},[487],[344,1938,1940,1944,1947],{"className":1939},[379],[344,1941],{"className":1942,"style":1943},[383],"height:0.8389em;vertical-align:-0.1944em;",[344,1945,1096],{"className":1946},[388],[344,1948,831],{"className":1949},[937],[15,1951,1952,1953,1981],{},"则称该点为 ",[344,1954,1956,1969],{"className":1955,"translate":348},[347],[344,1957,1959],{"className":1958},[352],[354,1960,1961],{"xmlns":356},[358,1962,1963,1967],{},[361,1964,1965],{},[364,1966,448],{},[368,1968,448],{"encoding":370},[344,1970,1972],{"className":1971,"ariaHidden":375},[374],[344,1973,1975,1978],{"className":1974},[379],[344,1976],{"className":1977,"style":479},[383],[344,1979,448],{"className":1980,"style":483},[388,389]," 的临界点。对开区域上的光滑函数，内部的局部极小值与局部极大值都必然是临界点。",[764,1983,1984,2156,2686,2771,3259],{},[15,1985,1986,702,1989,800,2017,2099,2100,2155],{},[324,1987,1988],{},"定义 2.3（Hessian）",[344,1990,1992,2005],{"className":1991,"translate":348},[347],[344,1993,1995],{"className":1994},[352],[354,1996,1997],{"xmlns":356},[358,1998,1999,2003],{},[361,2000,2001],{},[364,2002,448],{},[368,2004,448],{"encoding":370},[344,2006,2008],{"className":2007,"ariaHidden":375},[374],[344,2009,2011,2014],{"className":2010},[379],[344,2012],{"className":2013,"style":479},[383],[344,2015,448],{"className":2016,"style":483},[388,389],[344,2018,2020,2042],{"className":2019,"translate":348},[347],[344,2021,2023],{"className":2022},[352],[354,2024,2025],{"xmlns":356},[358,2026,2027,2039],{},[361,2028,2029,2031,2033],{},[364,2030,815],{},[450,2032,580],{},[454,2034,2035,2037],{},[364,2036,459],{"mathvariant":458},[364,2038,366],{},[368,2040,2041],{"encoding":370},"x\\in\\mathbb{R}^N",[344,2043,2045,2064],{"className":2044,"ariaHidden":375},[374],[344,2046,2048,2052,2055,2058,2061],{"className":2047},[379],[344,2049],{"className":2050,"style":2051},[383],"height:0.5782em;vertical-align:-0.0391em;",[344,2053,815],{"className":2054},[388,389],[344,2056],{"className":2057,"style":488},[487],[344,2059,580],{"className":2060},[492],[344,2062],{"className":2063,"style":488},[487],[344,2065,2067,2070],{"className":2066},[379],[344,2068],{"className":2069,"style":502},[383],[344,2071,2073,2076],{"className":2072},[388],[344,2074,459],{"className":2075},[388,509],[344,2077,2079],{"className":2078},[513],[344,2080,2082],{"className":2081},[517],[344,2083,2085],{"className":2084},[521],[344,2086,2088],{"className":2087,"style":502},[525],[344,2089,2090,2093],{"style":528},[344,2091],{"className":2092,"style":533},[532],[344,2094,2096],{"className":2095},[537,538,539,540],[344,2097,366],{"className":2098,"style":390},[388,389,540]," 处的 Hessian 是 ",[344,2101,2103,2122],{"className":2102,"translate":348},[347],[344,2104,2106],{"className":2105},[352],[354,2107,2108],{"xmlns":356},[358,2109,2110,2119],{},[361,2111,2112,2114,2117],{},[364,2113,366],{},[450,2115,2116],{},"×",[364,2118,366],{},[368,2120,2121],{"encoding":370},"N\\times N",[344,2123,2125,2146],{"className":2124,"ariaHidden":375},[374],[344,2126,2128,2132,2135,2139,2143],{"className":2127},[379],[344,2129],{"className":2130,"style":2131},[383],"height:0.7667em;vertical-align:-0.0833em;",[344,2133,366],{"className":2134,"style":390},[388,389],[344,2136],{"className":2137,"style":2138},[487],"margin-right:0.2222em;",[344,2140,2116],{"className":2141},[2142],"mbin",[344,2144],{"className":2145,"style":2138},[487],[344,2147,2149,2152],{"className":2148},[379],[344,2150],{"className":2151,"style":384},[383],[344,2153,366],{"className":2154,"style":390},[388,389]," 矩阵",[344,2157,2159],{"className":2158,"translate":348},[1047],[344,2160,2162,2272],{"className":2161,"translate":348},[347],[344,2163,2165],{"className":2164},[352],[354,2166,2167],{"xmlns":356,"display":1056},[358,2168,2169,2269],{},[361,2170,2171,2178,2180,2182,2184,2186,2192,2194,2196,2198,2200,2202,2204,2267],{},[822,2172,2173,2176],{},[364,2174,2175],{},"H",[364,2177,448],{},[450,2179,593],{"stretchy":592},[364,2181,815],{},[450,2183,602],{"stretchy":592},[450,2185,818],{},[454,2187,2188,2190],{},[364,2189,1064],{"mathvariant":1063},[587,2191,589],{},[364,2193,448],{},[450,2195,593],{"stretchy":592},[364,2197,815],{},[450,2199,602],{"stretchy":592},[450,2201,452],{},[450,2203,818],{},[2205,2206,2207,2253,2265],"msubsup",{},[361,2208,2209,2212,2244,2246,2248,2250],{},[450,2210,2211],{"fence":375},"[",[1102,2213,2214,2224],{},[361,2215,2216,2222],{},[454,2217,2218,2220],{},[364,2219,1108],{"mathvariant":1063},[587,2221,589],{},[364,2223,448],{},[361,2225,2226,2228,2235,2237],{},[364,2227,1108],{"mathvariant":1063},[822,2229,2230,2232],{},[364,2231,815],{},[364,2233,2234],{},"i",[364,2236,1108],{"mathvariant":1063},[822,2238,2239,2241],{},[364,2240,815],{},[364,2242,2243],{},"j",[450,2245,593],{"stretchy":592},[364,2247,815],{},[450,2249,602],{"stretchy":592},[450,2251,2252],{"fence":375},"]",[361,2254,2255,2257,2259,2261,2263],{},[364,2256,2234],{},[450,2258,831],{"separator":375},[364,2260,2243],{},[450,2262,818],{},[587,2264,828],{},[364,2266,366],{},[364,2268,1192],{"mathvariant":1063},[368,2270,2271],{"encoding":370},"H_f(x) = \\nabla^2 f(x) := \\left[\\frac{\\partial^2 f}{\\partial x_i \\partial x_j}(x)\\right]_{i,j=1}^N.",[344,2273,2275,2344,2402],{"className":2274,"ariaHidden":375},[374],[344,2276,2278,2282,2326,2329,2332,2335,2338,2341],{"className":2277},[379],[344,2279],{"className":2280,"style":2281},[383],"height:1.0361em;vertical-align:-0.2861em;",[344,2283,2285,2289],{"className":2284},[388],[344,2286,2175],{"className":2287,"style":2288},[388,389],"margin-right:0.0813em;",[344,2290,2292],{"className":2291},[513],[344,2293,2295,2317],{"className":2294},[517,900],[344,2296,2298,2314],{"className":2297},[521],[344,2299,2302],{"className":2300,"style":2301},[525],"height:0.3361em;",[344,2303,2305,2308],{"style":2304},"top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;",[344,2306],{"className":2307,"style":533},[532],[344,2309,2311],{"className":2310},[537,538,539,540],[344,2312,448],{"className":2313,"style":483},[388,389,540],[344,2315,924],{"className":2316},[923],[344,2318,2320],{"className":2319},[521],[344,2321,2324],{"className":2322,"style":2323},[525],"height:0.2861em;",[344,2325],{},[344,2327,593],{"className":2328},[668],[344,2330,815],{"className":2331},[388,389],[344,2333,602],{"className":2334},[701],[344,2336],{"className":2337,"style":488},[487],[344,2339,818],{"className":2340},[492],[344,2342],{"className":2343,"style":488},[487],[344,2345,2347,2351,2381,2384,2387,2390,2393,2396,2399],{"className":2346},[379],[344,2348],{"className":2349,"style":2350},[383],"height:1.1141em;vertical-align:-0.25em;",[344,2352,2354,2357],{"className":2353},[388],[344,2355,1064],{"className":2356},[388],[344,2358,2360],{"className":2359},[513],[344,2361,2363],{"className":2362},[517],[344,2364,2366],{"className":2365},[521],[344,2367,2370],{"className":2368,"style":2369},[525],"height:0.8641em;",[344,2371,2372,2375],{"style":1682},[344,2373],{"className":2374,"style":533},[532],[344,2376,2378],{"className":2377},[537,538,539,540],[344,2379,589],{"className":2380},[388,540],[344,2382,448],{"className":2383,"style":483},[388,389],[344,2385,593],{"className":2386},[668],[344,2388,815],{"className":2389},[388,389],[344,2391,602],{"className":2392},[701],[344,2394],{"className":2395,"style":488},[487],[344,2397,1226],{"className":2398},[492],[344,2400],{"className":2401,"style":488},[487],[344,2403,2405,2409,2680,2683],{"className":2404},[379],[344,2406],{"className":2407,"style":2408},[383],"height:2.8803em;vertical-align:-1.1579em;",[344,2410,2412,2616],{"className":2411},[945],[344,2413,2415,2423,2601,2604,2607,2610],{"className":2414},[945],[344,2416,2420],{"className":2417,"style":2419},[668,2418],"delimcenter","top:0em;",[344,2421,2211],{"className":2422},[1246,539],[344,2424,2426,2429,2598],{"className":2425},[388],[344,2427],{"className":2428},[668,1332],[344,2430,2432],{"className":2431},[1102],[344,2433,2435,2589],{"className":2434},[517,900],[344,2436,2438,2586],{"className":2437},[521],[344,2439,2442,2538,2546],{"className":2440,"style":2441},[525],"height:1.4911em;",[344,2443,2444,2447],{"style":1348},[344,2445],{"className":2446,"style":1352},[532],[344,2448,2450,2453,2494,2497],{"className":2449},[388],[344,2451,1108],{"className":2452,"style":1359},[388],[344,2454,2456,2459],{"className":2455},[388],[344,2457,815],{"className":2458},[388,389],[344,2460,2462],{"className":2461},[513],[344,2463,2465,2486],{"className":2464},[517,900],[344,2466,2468,2483],{"className":2467},[521],[344,2469,2472],{"className":2470,"style":2471},[525],"height:0.3117em;",[344,2473,2474,2477],{"style":910},[344,2475],{"className":2476,"style":533},[532],[344,2478,2480],{"className":2479},[537,538,539,540],[344,2481,2234],{"className":2482},[388,389,540],[344,2484,924],{"className":2485},[923],[344,2487,2489],{"className":2488},[521],[344,2490,2492],{"className":2491,"style":931},[525],[344,2493],{},[344,2495,1108],{"className":2496,"style":1359},[388],[344,2498,2500,2503],{"className":2499},[388],[344,2501,815],{"className":2502},[388,389],[344,2504,2506],{"className":2505},[513],[344,2507,2509,2530],{"className":2508},[517,900],[344,2510,2512,2527],{"className":2511},[521],[344,2513,2515],{"className":2514,"style":2471},[525],[344,2516,2517,2520],{"style":910},[344,2518],{"className":2519,"style":533},[532],[344,2521,2523],{"className":2522},[537,538,539,540],[344,2524,2243],{"className":2525,"style":2526},[388,389,540],"margin-right:0.0572em;",[344,2528,924],{"className":2529},[923],[344,2531,2533],{"className":2532},[521],[344,2534,2536],{"className":2535,"style":2323},[525],[344,2537],{},[344,2539,2540,2543],{"style":1402},[344,2541],{"className":2542,"style":1352},[532],[344,2544],{"className":2545,"style":1410},[1409],[344,2547,2548,2551],{"style":1413},[344,2549],{"className":2550,"style":1352},[532],[344,2552,2554,2583],{"className":2553},[388],[344,2555,2557,2560],{"className":2556},[388],[344,2558,1108],{"className":2559,"style":1359},[388],[344,2561,2563],{"className":2562},[513],[344,2564,2566],{"className":2565},[517],[344,2567,2569],{"className":2568},[521],[344,2570,2572],{"className":2571,"style":653},[525],[344,2573,2574,2577],{"style":528},[344,2575],{"className":2576,"style":533},[532],[344,2578,2580],{"className":2579},[537,538,539,540],[344,2581,589],{"className":2582},[388,540],[344,2584,448],{"className":2585,"style":483},[388,389],[344,2587,924],{"className":2588},[923],[344,2590,2592],{"className":2591},[521],[344,2593,2596],{"className":2594,"style":2595},[525],"height:0.9721em;",[344,2597],{},[344,2599],{"className":2600},[701,1332],[344,2602,593],{"className":2603},[668],[344,2605,815],{"className":2606},[388,389],[344,2608,602],{"className":2609},[701],[344,2611,2613],{"className":2612,"style":2419},[701,2418],[344,2614,2252],{"className":2615},[1246,539],[344,2617,2619],{"className":2618},[513],[344,2620,2622,2671],{"className":2621},[517,900],[344,2623,2625,2668],{"className":2624},[521],[344,2626,2629,2656],{"className":2627,"style":2628},[525],"height:1.7223em;",[344,2630,2632,2635],{"style":2631},"top:-1.6782em;margin-right:0.05em;",[344,2633],{"className":2634,"style":533},[532],[344,2636,2638],{"className":2637},[537,538,539,540],[344,2639,2641,2644,2647,2650,2653],{"className":2640},[388,540],[344,2642,2234],{"className":2643},[388,389,540],[344,2645,831],{"className":2646},[937,540],[344,2648,2243],{"className":2649,"style":2526},[388,389,540],[344,2651,818],{"className":2652},[492,540],[344,2654,828],{"className":2655},[388,540],[344,2657,2659,2662],{"style":2658},"top:-3.944em;margin-right:0.05em;",[344,2660],{"className":2661,"style":533},[532],[344,2663,2665],{"className":2664},[537,538,539,540],[344,2666,366],{"className":2667,"style":390},[388,389,540],[344,2669,924],{"className":2670},[923],[344,2672,2674],{"className":2673},[521],[344,2675,2678],{"className":2676,"style":2677},[525],"height:1.1579em;",[344,2679],{},[344,2681],{"className":2682,"style":941},[487],[344,2684,1192],{"className":2685},[388],[15,2687,2688,2689,2770],{},"由于 ",[344,2690,2692,2714],{"className":2691,"translate":348},[347],[344,2693,2695],{"className":2694},[352],[354,2696,2697],{"xmlns":356},[358,2698,2699,2711],{},[361,2700,2701,2703,2705],{},[364,2702,448],{},[450,2704,580],{},[454,2706,2707,2709],{},[364,2708,585],{},[587,2710,589],{},[368,2712,2713],{"encoding":370},"f\\in C^2",[344,2715,2717,2735],{"className":2716,"ariaHidden":375},[374],[344,2718,2720,2723,2726,2729,2732],{"className":2719},[379],[344,2721],{"className":2722,"style":479},[383],[344,2724,448],{"className":2725,"style":483},[388,389],[344,2727],{"className":2728,"style":488},[487],[344,2730,580],{"className":2731},[492],[344,2733],{"className":2734,"style":488},[487],[344,2736,2738,2741],{"className":2737},[379],[344,2739],{"className":2740,"style":653},[383],[344,2742,2744,2747],{"className":2743},[388],[344,2745,585],{"className":2746,"style":640},[388,389],[344,2748,2750],{"className":2749},[513],[344,2751,2753],{"className":2752},[517],[344,2754,2756],{"className":2755},[521],[344,2757,2759],{"className":2758,"style":653},[525],[344,2760,2761,2764],{"style":528},[344,2762],{"className":2763,"style":533},[532],[344,2765,2767],{"className":2766},[537,538,539,540],[344,2768,589],{"className":2769},[388,540],"，由 Clairaut 定理可得",[344,2772,2774],{"className":2773,"translate":348},[1047],[344,2775,2777,2865],{"className":2776,"translate":348},[347],[344,2778,2780],{"className":2779},[352],[354,2781,2782],{"xmlns":356,"display":1056},[358,2783,2784,2862],{},[361,2785,2786,2816,2818,2820,2822,2824,2854,2856,2858,2860],{},[1102,2787,2788,2798],{},[361,2789,2790,2796],{},[454,2791,2792,2794],{},[364,2793,1108],{"mathvariant":1063},[587,2795,589],{},[364,2797,448],{},[361,2799,2800,2802,2808,2810],{},[364,2801,1108],{"mathvariant":1063},[822,2803,2804,2806],{},[364,2805,815],{},[364,2807,2234],{},[364,2809,1108],{"mathvariant":1063},[822,2811,2812,2814],{},[364,2813,815],{},[364,2815,2243],{},[450,2817,593],{"stretchy":592},[364,2819,815],{},[450,2821,602],{"stretchy":592},[450,2823,818],{},[1102,2825,2826,2836],{},[361,2827,2828,2834],{},[454,2829,2830,2832],{},[364,2831,1108],{"mathvariant":1063},[587,2833,589],{},[364,2835,448],{},[361,2837,2838,2840,2846,2848],{},[364,2839,1108],{"mathvariant":1063},[822,2841,2842,2844],{},[364,2843,815],{},[364,2845,2243],{},[364,2847,1108],{"mathvariant":1063},[822,2849,2850,2852],{},[364,2851,815],{},[364,2853,2234],{},[450,2855,593],{"stretchy":592},[364,2857,815],{},[450,2859,602],{"stretchy":592},[450,2861,831],{"separator":375},[368,2863,2864],{"encoding":370},"\\frac{\\partial^2 f}{\\partial x_i \\partial x_j}(x) = \\frac{\\partial^2 f}{\\partial x_j \\partial x_i}(x),",[344,2866,2868,3067],{"className":2867,"ariaHidden":375},[374],[344,2869,2871,2875,3049,3052,3055,3058,3061,3064],{"className":2870},[379],[344,2872],{"className":2873,"style":2874},[383],"height:2.4632em;vertical-align:-0.9721em;",[344,2876,2878,2881,3046],{"className":2877},[388],[344,2879],{"className":2880},[668,1332],[344,2882,2884],{"className":2883},[1102],[344,2885,2887,3038],{"className":2886},[517,900],[344,2888,2890,3035],{"className":2889},[521],[344,2891,2893,2987,2995],{"className":2892,"style":2441},[525],[344,2894,2895,2898],{"style":1348},[344,2896],{"className":2897,"style":1352},[532],[344,2899,2901,2904,2944,2947],{"className":2900},[388],[344,2902,1108],{"className":2903,"style":1359},[388],[344,2905,2907,2910],{"className":2906},[388],[344,2908,815],{"className":2909},[388,389],[344,2911,2913],{"className":2912},[513],[344,2914,2916,2936],{"className":2915},[517,900],[344,2917,2919,2933],{"className":2918},[521],[344,2920,2922],{"className":2921,"style":2471},[525],[344,2923,2924,2927],{"style":910},[344,2925],{"className":2926,"style":533},[532],[344,2928,2930],{"className":2929},[537,538,539,540],[344,2931,2234],{"className":2932},[388,389,540],[344,2934,924],{"className":2935},[923],[344,2937,2939],{"className":2938},[521],[344,2940,2942],{"className":2941,"style":931},[525],[344,2943],{},[344,2945,1108],{"className":2946,"style":1359},[388],[344,2948,2950,2953],{"className":2949},[388],[344,2951,815],{"className":2952},[388,389],[344,2954,2956],{"className":2955},[513],[344,2957,2959,2979],{"className":2958},[517,900],[344,2960,2962,2976],{"className":2961},[521],[344,2963,2965],{"className":2964,"style":2471},[525],[344,2966,2967,2970],{"style":910},[344,2968],{"className":2969,"style":533},[532],[344,2971,2973],{"className":2972},[537,538,539,540],[344,2974,2243],{"className":2975,"style":2526},[388,389,540],[344,2977,924],{"className":2978},[923],[344,2980,2982],{"className":2981},[521],[344,2983,2985],{"className":2984,"style":2323},[525],[344,2986],{},[344,2988,2989,2992],{"style":1402},[344,2990],{"className":2991,"style":1352},[532],[344,2993],{"className":2994,"style":1410},[1409],[344,2996,2997,3000],{"style":1413},[344,2998],{"className":2999,"style":1352},[532],[344,3001,3003,3032],{"className":3002},[388],[344,3004,3006,3009],{"className":3005},[388],[344,3007,1108],{"className":3008,"style":1359},[388],[344,3010,3012],{"className":3011},[513],[344,3013,3015],{"className":3014},[517],[344,3016,3018],{"className":3017},[521],[344,3019,3021],{"className":3020,"style":653},[525],[344,3022,3023,3026],{"style":528},[344,3024],{"className":3025,"style":533},[532],[344,3027,3029],{"className":3028},[537,538,539,540],[344,3030,589],{"className":3031},[388,540],[344,3033,448],{"className":3034,"style":483},[388,389],[344,3036,924],{"className":3037},[923],[344,3039,3041],{"className":3040},[521],[344,3042,3044],{"className":3043,"style":2595},[525],[344,3045],{},[344,3047],{"className":3048},[701,1332],[344,3050,593],{"className":3051},[668],[344,3053,815],{"className":3054},[388,389],[344,3056,602],{"className":3057},[701],[344,3059],{"className":3060,"style":488},[487],[344,3062,818],{"className":3063},[492],[344,3065],{"className":3066,"style":488},[487],[344,3068,3070,3073,3247,3250,3253,3256],{"className":3069},[379],[344,3071],{"className":3072,"style":2874},[383],[344,3074,3076,3079,3244],{"className":3075},[388],[344,3077],{"className":3078},[668,1332],[344,3080,3082],{"className":3081},[1102],[344,3083,3085,3236],{"className":3084},[517,900],[344,3086,3088,3233],{"className":3087},[521],[344,3089,3091,3185,3193],{"className":3090,"style":2441},[525],[344,3092,3093,3096],{"style":1348},[344,3094],{"className":3095,"style":1352},[532],[344,3097,3099,3102,3142,3145],{"className":3098},[388],[344,3100,1108],{"className":3101,"style":1359},[388],[344,3103,3105,3108],{"className":3104},[388],[344,3106,815],{"className":3107},[388,389],[344,3109,3111],{"className":3110},[513],[344,3112,3114,3134],{"className":3113},[517,900],[344,3115,3117,3131],{"className":3116},[521],[344,3118,3120],{"className":3119,"style":2471},[525],[344,3121,3122,3125],{"style":910},[344,3123],{"className":3124,"style":533},[532],[344,3126,3128],{"className":3127},[537,538,539,540],[344,3129,2243],{"className":3130,"style":2526},[388,389,540],[344,3132,924],{"className":3133},[923],[344,3135,3137],{"className":3136},[521],[344,3138,3140],{"className":3139,"style":2323},[525],[344,3141],{},[344,3143,1108],{"className":3144,"style":1359},[388],[344,3146,3148,3151],{"className":3147},[388],[344,3149,815],{"className":3150},[388,389],[344,3152,3154],{"className":3153},[513],[344,3155,3157,3177],{"className":3156},[517,900],[344,3158,3160,3174],{"className":3159},[521],[344,3161,3163],{"className":3162,"style":2471},[525],[344,3164,3165,3168],{"style":910},[344,3166],{"className":3167,"style":533},[532],[344,3169,3171],{"className":3170},[537,538,539,540],[344,3172,2234],{"className":3173},[388,389,540],[344,3175,924],{"className":3176},[923],[344,3178,3180],{"className":3179},[521],[344,3181,3183],{"className":3182,"style":931},[525],[344,3184],{},[344,3186,3187,3190],{"style":1402},[344,3188],{"className":3189,"style":1352},[532],[344,3191],{"className":3192,"style":1410},[1409],[344,3194,3195,3198],{"style":1413},[344,3196],{"className":3197,"style":1352},[532],[344,3199,3201,3230],{"className":3200},[388],[344,3202,3204,3207],{"className":3203},[388],[344,3205,1108],{"className":3206,"style":1359},[388],[344,3208,3210],{"className":3209},[513],[344,3211,3213],{"className":3212},[517],[344,3214,3216],{"className":3215},[521],[344,3217,3219],{"className":3218,"style":653},[525],[344,3220,3221,3224],{"style":528},[344,3222],{"className":3223,"style":533},[532],[344,3225,3227],{"className":3226},[537,538,539,540],[344,3228,589],{"className":3229},[388,540],[344,3231,448],{"className":3232,"style":483},[388,389],[344,3234,924],{"className":3235},[923],[344,3237,3239],{"className":3238},[521],[344,3240,3242],{"className":3241,"style":2595},[525],[344,3243],{},[344,3245],{"className":3246},[701,1332],[344,3248,593],{"className":3249},[668],[344,3251,815],{"className":3252},[388,389],[344,3254,602],{"className":3255},[701],[344,3257,831],{"className":3258},[937],[15,3260,3261,3262,3347],{},"因此 ",[344,3263,3265,3289],{"className":3264,"translate":348},[347],[344,3266,3268],{"className":3267},[352],[354,3269,3270],{"xmlns":356},[358,3271,3272,3286],{},[361,3273,3274,3280,3282,3284],{},[822,3275,3276,3278],{},[364,3277,2175],{},[364,3279,448],{},[450,3281,593],{"stretchy":592},[364,3283,815],{},[450,3285,602],{"stretchy":592},[368,3287,3288],{"encoding":370},"H_f(x)",[344,3290,3292],{"className":3291,"ariaHidden":375},[374],[344,3293,3295,3298,3338,3341,3344],{"className":3294},[379],[344,3296],{"className":3297,"style":2281},[383],[344,3299,3301,3304],{"className":3300},[388],[344,3302,2175],{"className":3303,"style":2288},[388,389],[344,3305,3307],{"className":3306},[513],[344,3308,3310,3330],{"className":3309},[517,900],[344,3311,3313,3327],{"className":3312},[521],[344,3314,3316],{"className":3315,"style":2301},[525],[344,3317,3318,3321],{"style":2304},[344,3319],{"className":3320,"style":533},[532],[344,3322,3324],{"className":3323},[537,538,539,540],[344,3325,448],{"className":3326,"style":483},[388,389,540],[344,3328,924],{"className":3329},[923],[344,3331,3333],{"className":3332},[521],[344,3334,3336],{"className":3335,"style":2323},[525],[344,3337],{},[344,3339,593],{"className":3340},[668],[344,3342,815],{"className":3343},[388,389],[344,3345,602],{"className":3346},[701]," 是实对称矩阵。",[15,3349,3350,3351,3470],{},"Hessian 之所以重要，因为它是泰勒展开中的二次项。设 ",[344,3352,3354,3382],{"className":3353,"translate":348},[347],[344,3355,3357],{"className":3356},[352],[354,3358,3359],{"xmlns":356},[358,3360,3361,3379],{},[361,3362,3363,3366,3368,3370,3373],{},[364,3364,3365],{},"h",[450,3367,818],{},[364,3369,815],{},[450,3371,3372],{},"−",[822,3374,3375,3377],{},[364,3376,815],{},[587,3378,1096],{},[368,3380,3381],{"encoding":370},"h = x - x_0",[344,3383,3385,3404,3423],{"className":3384,"ariaHidden":375},[374],[344,3386,3388,3392,3395,3398,3401],{"className":3387},[379],[344,3389],{"className":3390,"style":3391},[383],"height:0.6944em;",[344,3393,3365],{"className":3394},[388,389],[344,3396],{"className":3397,"style":488},[487],[344,3399,818],{"className":3400},[492],[344,3402],{"className":3403,"style":488},[487],[344,3405,3407,3411,3414,3417,3420],{"className":3406},[379],[344,3408],{"className":3409,"style":3410},[383],"height:0.6667em;vertical-align:-0.0833em;",[344,3412,815],{"className":3413},[388,389],[344,3415],{"className":3416,"style":2138},[487],[344,3418,3372],{"className":3419},[2142],[344,3421],{"className":3422,"style":2138},[487],[344,3424,3426,3430],{"className":3425},[379],[344,3427],{"className":3428,"style":3429},[383],"height:0.5806em;vertical-align:-0.15em;",[344,3431,3433,3436],{"className":3432},[388],[344,3434,815],{"className":3435},[388,389],[344,3437,3439],{"className":3438},[513],[344,3440,3442,3462],{"className":3441},[517,900],[344,3443,3445,3459],{"className":3444},[521],[344,3446,3448],{"className":3447,"style":907},[525],[344,3449,3450,3453],{"style":910},[344,3451],{"className":3452,"style":533},[532],[344,3454,3456],{"className":3455},[537,538,539,540],[344,3457,1096],{"className":3458},[388,540],[344,3460,924],{"className":3461},[923],[344,3463,3465],{"className":3464},[521],[344,3466,3468],{"className":3467,"style":931},[525],[344,3469],{},"，则",[344,3472,3474],{"className":3473,"translate":348},[1047],[344,3475,3477,3615],{"className":3476,"translate":348},[347],[344,3478,3480],{"className":3479},[352],[354,3481,3482],{"xmlns":356,"display":1056},[358,3483,3484,3612],{},[1082,3485,3487],{"width":3486},"100%",[1088,3488,3489,3492,3604,3606],{},[1091,3490],{"width":3491},"50%",[1091,3493,3494],{},[361,3495,3496,3498,3500,3506,3509,3511,3513,3515,3517,3519,3525,3527,3529,3531,3533,3535,3541,3548,3550,3552,3558,3564,3570,3572,3578,3580,3582,3584,3587,3589,3592,3594,3600,3602],{},[364,3497,448],{},[450,3499,593],{"stretchy":592},[822,3501,3502,3504],{},[364,3503,815],{},[587,3505,1096],{},[450,3507,3508],{},"+",[364,3510,3365],{},[450,3512,602],{"stretchy":592},[450,3514,818],{},[364,3516,448],{},[450,3518,593],{"stretchy":592},[822,3520,3521,3523],{},[364,3522,815],{},[587,3524,1096],{},[450,3526,602],{"stretchy":592},[450,3528,3508],{},[364,3530,1064],{"mathvariant":1063},[364,3532,448],{},[450,3534,593],{"stretchy":592},[822,3536,3537,3539],{},[364,3538,815],{},[587,3540,1096],{},[454,3542,3543,3545],{},[450,3544,602],{"stretchy":592},[364,3546,3547],{},"T",[364,3549,3365],{},[450,3551,3508],{},[1102,3553,3554,3556],{},[587,3555,828],{},[587,3557,589],{},[454,3559,3560,3562],{},[364,3561,3365],{},[364,3563,3547],{},[822,3565,3566,3568],{},[364,3567,2175],{},[364,3569,448],{},[450,3571,593],{"stretchy":592},[822,3573,3574,3576],{},[364,3575,815],{},[587,3577,1096],{},[450,3579,602],{"stretchy":592},[364,3581,3365],{},[450,3583,3508],{},[364,3585,3586],{},"o",[450,3588,593],{"stretchy":592},[364,3590,3591],{"mathvariant":1063},"∥",[364,3593,3365],{},[454,3595,3596,3598],{},[364,3597,3591],{"mathvariant":1063},[587,3599,589],{},[450,3601,602],{"stretchy":592},[364,3603,1192],{"mathvariant":1063},[1091,3605],{"width":3491},[1091,3607,3608],{},[3609,3610,3611],"mtext",{},"(1)",[368,3613,3614],{"encoding":370},"f(x_0+h) = f(x_0) + \\nabla f(x_0)^T h + \\frac{1}{2}h^T H_f(x_0) h + o(\\|h\\|^2). \\tag{1}",[344,3616,3618,3679,3700,3764,3862,4060,4113],{"className":3617,"ariaHidden":375},[374],[344,3619,3621,3624,3627,3630,3670,3673,3676],{"className":3620},[379],[344,3622],{"className":3623,"style":884},[383],[344,3625,448],{"className":3626,"style":483},[388,389],[344,3628,593],{"className":3629},[668],[344,3631,3633,3636],{"className":3632},[388],[344,3634,815],{"className":3635},[388,389],[344,3637,3639],{"className":3638},[513],[344,3640,3642,3662],{"className":3641},[517,900],[344,3643,3645,3659],{"className":3644},[521],[344,3646,3648],{"className":3647,"style":907},[525],[344,3649,3650,3653],{"style":910},[344,3651],{"className":3652,"style":533},[532],[344,3654,3656],{"className":3655},[537,538,539,540],[344,3657,1096],{"className":3658},[388,540],[344,3660,924],{"className":3661},[923],[344,3663,3665],{"className":3664},[521],[344,3666,3668],{"className":3667,"style":931},[525],[344,3669],{},[344,3671],{"className":3672,"style":2138},[487],[344,3674,3508],{"className":3675},[2142],[344,3677],{"className":3678,"style":2138},[487],[344,3680,3682,3685,3688,3691,3694,3697],{"className":3681},[379],[344,3683],{"className":3684,"style":884},[383],[344,3686,3365],{"className":3687},[388,389],[344,3689,602],{"className":3690},[701],[344,3692],{"className":3693,"style":488},[487],[344,3695,818],{"className":3696},[492],[344,3698],{"className":3699,"style":488},[487],[344,3701,3703,3706,3709,3712,3752,3755,3758,3761],{"className":3702},[379],[344,3704],{"className":3705,"style":884},[383],[344,3707,448],{"className":3708,"style":483},[388,389],[344,3710,593],{"className":3711},[668],[344,3713,3715,3718],{"className":3714},[388],[344,3716,815],{"className":3717},[388,389],[344,3719,3721],{"className":3720},[513],[344,3722,3724,3744],{"className":3723},[517,900],[344,3725,3727,3741],{"className":3726},[521],[344,3728,3730],{"className":3729,"style":907},[525],[344,3731,3732,3735],{"style":910},[344,3733],{"className":3734,"style":533},[532],[344,3736,3738],{"className":3737},[537,538,539,540],[344,3739,1096],{"className":3740},[388,540],[344,3742,924],{"className":3743},[923],[344,3745,3747],{"className":3746},[521],[344,3748,3750],{"className":3749,"style":931},[525],[344,3751],{},[344,3753,602],{"className":3754},[701],[344,3756],{"className":3757,"style":2138},[487],[344,3759,3508],{"className":3760},[2142],[344,3762],{"className":3763,"style":2138},[487],[344,3765,3767,3771,3774,3777,3780,3820,3850,3853,3856,3859],{"className":3766},[379],[344,3768],{"className":3769,"style":3770},[383],"height:1.1413em;vertical-align:-0.25em;",[344,3772,1064],{"className":3773},[388],[344,3775,448],{"className":3776,"style":483},[388,389],[344,3778,593],{"className":3779},[668],[344,3781,3783,3786],{"className":3782},[388],[344,3784,815],{"className":3785},[388,389],[344,3787,3789],{"className":3788},[513],[344,3790,3792,3812],{"className":3791},[517,900],[344,3793,3795,3809],{"className":3794},[521],[344,3796,3798],{"className":3797,"style":907},[525],[344,3799,3800,3803],{"style":910},[344,3801],{"className":3802,"style":533},[532],[344,3804,3806],{"className":3805},[537,538,539,540],[344,3807,1096],{"className":3808},[388,540],[344,3810,924],{"className":3811},[923],[344,3813,3815],{"className":3814},[521],[344,3816,3818],{"className":3817,"style":931},[525],[344,3819],{},[344,3821,3823,3826],{"className":3822},[701],[344,3824,602],{"className":3825},[701],[344,3827,3829],{"className":3828},[513],[344,3830,3832],{"className":3831},[517],[344,3833,3835],{"className":3834},[521],[344,3836,3838],{"className":3837,"style":1661},[525],[344,3839,3840,3843],{"style":1682},[344,3841],{"className":3842,"style":533},[532],[344,3844,3846],{"className":3845},[537,538,539,540],[344,3847,3547],{"className":3848,"style":3849},[388,389,540],"margin-right:0.1389em;",[344,3851,3365],{"className":3852},[388,389],[344,3854],{"className":3855,"style":2138},[487],[344,3857,3508],{"className":3858},[2142],[344,3860],{"className":3861,"style":2138},[487],[344,3863,3865,3869,3933,3962,4002,4005,4045,4048,4051,4054,4057],{"className":3864},[379],[344,3866],{"className":3867,"style":3868},[383],"height:2.0074em;vertical-align:-0.686em;",[344,3870,3872,3875,3930],{"className":3871},[388],[344,3873],{"className":3874},[668,1332],[344,3876,3878],{"className":3877},[1102],[344,3879,3881,3921],{"className":3880},[517,900],[344,3882,3884,3918],{"className":3883},[521],[344,3885,3888,3899,3907],{"className":3886,"style":3887},[525],"height:1.3214em;",[344,3889,3890,3893],{"style":1348},[344,3891],{"className":3892,"style":1352},[532],[344,3894,3896],{"className":3895},[388],[344,3897,589],{"className":3898},[388],[344,3900,3901,3904],{"style":1402},[344,3902],{"className":3903,"style":1352},[532],[344,3905],{"className":3906,"style":1410},[1409],[344,3908,3909,3912],{"style":1413},[344,3910],{"className":3911,"style":1352},[532],[344,3913,3915],{"className":3914},[388],[344,3916,828],{"className":3917},[388],[344,3919,924],{"className":3920},[9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",[344,4140,4142,4160],{"className":4141,"translate":348},[347],[344,4143,4145],{"className":4144},[352],[354,4146,4147],{"xmlns":356},[358,4148,4149,4157],{},[361,4150,4151],{},[822,4152,4153,4155],{},[364,4154,815],{},[364,4156,1721],{},[368,4158,4159],{"encoding":370},"x_c",[344,4161,4163],{"className":4162,"ariaHidden":375},[374],[344,4164,4166,4169],{"className":4165},[379],[344,4167],{"className":4168,"style":3429},[383],[344,4170,4172,4175],{"className":4171},[388],[344,4173,815],{"className":4174},[388,389],[344,4176,4178],{"className":4177},[513],[344,4179,4181,4201],{"className":4180},[517,900],[344,4182,4184,4198],{"className":4183},[521],[344,4185,4187],{"className":4186,"style":1761},[525],[344,4188,4189,4192],{"style":910},[344,4190],{"className":4191,"style":533},[532],[344,4193,4195],{"className":4194},[537,538,539,540],[344,4196,1721],{"className":4197},[388,389,540],[344,4199,924],{"className":4200},[923],[344,4202,4204],{"className":4203},[521],[344,4205,4207],{"className":4206,"style":931},[525],[344,4208],{}," 处，线性项消失，因此局部几何由下式决定到二阶：",[344,4211,4213],{"className":4212,"translate":348},[1047],[344,4214,4216,4310],{"className":4215,"translate":348},[347],[344,4217,4219],{"className":4218},[352],[354,4220,4221],{"xmlns":356,"display":1056},[358,4222,4223,4307],{},[361,4224,4225,4227,4229,4235,4237,4239,4241,4243,4245,4247,4253,4255,4257,4263,4269,4275,4277,4283,4285,4287,4289,4291,4293,4295,4297,4303,4305],{},[364,4226,448],{},[450,4228,593],{"stretchy":592},[822,4230,4231,4233],{},[364,4232,815],{},[364,4234,1721],{},[450,4236,3508],{},[364,4238,3365],{},[450,4240,602],{"stretchy":592},[450,4242,3372],{},[364,4244,448],{},[450,4246,593],{"stretchy":592},[822,4248,4249,4251],{},[364,4250,815],{},[364,4252,1721],{},[450,4254,602],{"stretchy":592},[450,4256,818],{},[1102,4258,4259,4261],{},[587,4260,828],{},[587,4262,589],{},[454,4264,4265,4267],{},[364,4266,3365],{},[364,4268,3547],{},[822,4270,4271,4273],{},[364,4272,2175],{},[364,4274,448],{},[450,4276,593],{"stretchy":592},[822,4278,4279,4281],{},[364,4280,815],{},[364,4282,1721],{},[450,4284,602],{"stretchy":592},[364,4286,3365],{},[450,4288,3508],{},[364,4290,3586],{},[450,4292,593],{"stretchy":592},[364,4294,3591],{"mathvariant":1063},[364,4296,3365],{},[454,4298,4299,4301],{},[364,4300,3591],{"mathvariant":1063},[587,4302,589],{},[450,4304,602],{"stretchy":592},[364,4306,1192],{"mathvariant":1063},[368,4308,4309],{"encoding":370},"f(x_c+h) - f(x_c) = \\frac{1}{2}h^T H_f(x_c)h + o(\\|h\\|^2).",[344,4311,4313,4374,4395,4459,4654],{"className":4312,"ariaHidden":375},[374],[344,4314,4316,4319,4322,4325,4365,4368,4371],{"className":4315},[379],[344,4317],{"className":4318,"style":884},[383],[344,4320,448],{"className":4321,"style":483},[388,389],[344,4323,593],{"className":4324},[668],[344,4326,4328,4331],{"className":4327},[388],[344,4329,815],{"className":4330},[388,389],[344,4332,4334],{"className":4333},[513],[344,4335,4337,4357],{"className":4336},[517,900],[344,4338,4340,4354],{"className":4339},[521],[344,4341,4343],{"className":4342,"style":1761},[525],[344,4344,4345,4348],{"style":910},[344,4346],{"className":4347,"style":533},[532],[344,4349,4351],{"className":4350},[537,538,539,540],[344,4352,1721],{"className":4353},[388,389,540],[344,4355,924],{"className":4356},[923],[344,4358,4360],{"className":4359},[521],[344,4361,4363],{"className":4362,"style":931},[525],[344,4364],{},[344,4366],{"className":4367,"style":2138},[487],[344,4369,3508],{"className":4370},[2142],[344,4372],{"className":4373,"style":2138},[487],[344,4375,4377,4380,4383,4386,4389,4392],{"className":4376},[379],[344,4378],{"className":4379,"style":884},[383],[344,4381,3365],{"className":4382},[388,389],[344,4384,602],{"className":4385},[701],[344,4387],{"className":4388,"style":2138},[487],[344,4390,3372],{"className":4391},[2142],[344,4393],{"className":4394,"style":2138},[487],[344,4396,4398,4401,4404,4407,4447,4450,4453,4456],{"className":4397},[379],[344,4399],{"className":4400,"style":884},[383],[344,4402,448],{"className":4403,"style":483},[388,389],[344,4405,593],{"className":4406},[668],[344,4408,4410,4413],{"className":4409},[388],[344,4411,815],{"className":4412},[388,389],[344,4414,4416],{"className":4415},[513],[344,4417,4419,4439],{"className":4418},[517,900],[344,4420,4422,4436],{"className":4421},[521],[344,4423,4425],{"className":4424,"style":1761},[525],[344,4426,4427,4430],{"style":910},[344,4428],{"className":4429,"style":533},[532],[344,4431,4433],{"className":4432},[537,538,539,540],[344,4434,1721],{"className":4435},[388,389,540],[344,4437,924],{"className":4438},[923],[344,4440,4442],{"className":4441},[521],[344,4443,4445],{"className":4444,"style":931},[525],[344,4446],{},[344,4448,602],{"className":4449},[701],[344,4451],{"className":4452,"style":488},[487],[344,4454,818],{"className":4455},[492],[344,4457],{"className":4458,"style":488},[487],[344,4460,4462,4465,4527,4556,4596,4599,4639,4642,4645,4648,4651],{"className":4461},[379],[344,4463],{"className":4464,"style":3868},[383],[344,4466,4468,4471,4524],{"className":4467},[388],[344,4469],{"className":4470},[668,1332],[344,4472,4474],{"className":4473},[1102],[344,4475,4477,4516],{"className":4476},[517,900],[344,4478,4480,4513],{"className":4479},[521],[344,4481,4483,4494,4502],{"className":4482,"style":3887},[525],[344,4484,4485,4488],{"style":1348},[344,4486],{"className":4487,"style":1352},[532],[344,4489,4491],{"className":4490},[388],[344,4492,589],{"className":4493},[388],[344,4495,4496,4499],{"style":1402},[344,4497],{"className":4498,"style":1352},[532],[344,4500],{"className":4501,"style":1410},[1409],[344,4503,4504,4507],{"style":1413},[344,4505],{"className":4506,"style":1352},[532],[344,4508,4510],{"className":4509},[388],[344,4511,828],{"className":4512},[388],[344,4514,924],{"className":4515},[923],[344,4517,4519],{"className":4518},[521],[344,4520,4522],{"className":4521,"style":3927},[525],[344,4523],{},[344,4525],{"className":4526},[701,1332],[344,4528,4530,4533],{"className":4529},[388],[344,4531,3365],{"className":4532},[388,389],[344,4534,4536],{"className":4535},[513],[344,4537,4539],{"className":4538},[517],[344,4540,4542],{"className":4541},[521],[344,4543,4545],{"className":4544,"style":1661},[525],[344,4546,4547,4550],{"style":1682},[344,4548],{"className":4549,"style":533},[532],[344,4551,4553],{"className":4552},[537,538,539,540],[344,4554,3547],{"className":4555,"style":3849},[388,389,540],[344,4557,4559,4562],{"className":4558},[388],[344,4560,2175],{"className":4561,"style":2288},[388,389],[344,4563,4565],{"className":4564},[513],[344,4566,4568,4588],{"className":4567},[517,900],[344,4569,4571,4585],{"className":4570},[521],[344,4572,4574],{"className":4573,"style":2301},[525],[344,4575,4576,4579],{"style":2304},[344,4577],{"className":4578,"style":533},[532],[344,4580,4582],{"className":4581},[537,538,539,540],[344,4583,448],{"className":4584,"style":483},[388,389,540],[344,4586,924],{"className":4587},[923],[344,4589,4591],{"className":4590},[521],[344,4592,4594],{"className":4593,"style":2323},[525],[344,4595],{},[344,4597,593],{"className":4598},[668],[344,4600,4602,4605],{"className":4601},[388],[344,4603,815],{"className":4604},[388,389],[344,4606,4608],{"className":4607},[513],[344,4609,4611,4631],{"className":4610},[517,900],[344,4612,4614,4628],{"className":4613},[521],[344,4615,4617],{"className":4616,"style":1761},[525],[344,4618,4619,4622],{"style":910},[344,4620],{"className":4621,"styl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是实对称矩阵，谱定理给出由特征向量 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组成的标准正交基，对应的实特征值为 ",[344,4982,4984,5015],{"className":4983,"translate":348},[347],[344,4985,4987],{"className":4986},[352],[354,4988,4989],{"xmlns":356},[358,4990,4991,5012],{},[361,4992,4993,5000,5002,5004,5006],{},[822,4994,4995,4998],{},[364,4996,4997],{},"λ",[587,4999,828],{},[450,5001,831],{"separator":375},[450,5003,834],{},[450,5005,831],{"separator":375},[822,5007,5008,5010],{},[364,5009,4997],{},[364,5011,366],{},[368,5013,5014],{"encoding":370},"\\lambda_1,\\dots,\\lambda_N",[344,5016,5018],{"className":5017,"ariaHidden":375},[374],[344,5019,5021,5024,5064,5067,5070,5073,5076,5079,5082],{"className":5020},[379],[344,5022],{"className":5023,"style":479},[383],[344,5025,5027,5030],{"className":5026},[388],[344,5028,4997],{"className":5029},[388,389],[344,5031,5033],{"className":5032},[513],[344,5034,5036,5056],{"className":5035},[517,900],[344,5037,5039,5053],{"className":5038},[521],[344,5040,5042],{"className":5041,"style":907},[525],[344,5043,5044,5047],{"style":910},[344,5045],{"className":5046,"style":533},[532],[344,5048,5050],{"className":5049},[537,538,539,540],[344,5051,828],{"className":5052},[388,540],[344,5054,924],{"className":5055},[923],[344,5057,5059],{"className":5058},[521],[344,5060,5062],{"className":5061,"style":931},[525],[344,5063],{},[344,5065,831],{"className":5066},[937],[344,5068],{"className":5069,"style":941},[487],[344,5071,834],{"className":5072},[945],[344,5074],{"className":5075,"style":941},[487],[344,5077,831],{"className":5078},[937],[344,5080],{"className":5081,"style":941},[487],[344,5083,5085,5088],{"className":5084},[388],[344,5086,4997],{"className":5087},[388,389],[344,5089,5091],{"className":5090},[513],[344,5092,5094,5114],{"className":5093},[517,900],[344,5095,5097,5111],{"className":5096},[521],[344,5098,5100],{"className":5099,"style":973},[525],[344,5101,5102,5105],{"style":910},[344,5103],{"className":5104,"style":533},[532],[344,5106,5108],{"className":5107},[537,538,539,540],[344,5109,366],{"className":5110,"style":390},[388,389,540],[344,5112,924],{"className":5113},[923],[344,5115,5117],{"className":5116},[521],[344,5118,5120],{"className":5119,"style":931},[525],[344,5121],{},"。若",[344,5124,5126],{"className":5125,"translate":348},[1047],[344,5127,5129,5176],{"className":5128,"translate":348},[347],[344,5130,5132],{"className":5131},[352],[354,5133,5134],{"xmlns":356,"display":1056},[358,5135,5136,5173],{},[361,5137,5138,5140,5142,5158,5165,5171],{},[364,5139,3365],{},[450,5141,818],{},[5143,5144,5145,5148,5156],"munderover",{},[450,5146,5147],{},"∑",[361,5149,5150,5152,5154],{},[364,5151,2234],{},[450,5153,818],{},[587,5155,828],{},[364,5157,366],{},[822,5159,5160,5163],{},[364,5161,5162],{},"a",[364,5164,2234],{},[822,5166,5167,5169],{},[364,5168,4852],{},[364,5170,2234],{},[450,5172,831],{"separator":375},[368,5174,5175],{"encoding":370},"h = \\sum_{i=1}^N a_i v_i,",[344,5177,5179,5197],{"className":5178,"ariaHidden":375},[374],[344,5180,5182,5185,5188,5191,5194],{"className":5181},[379],[344,5183],{"className":5184,"style":3391},[383],[344,5186,3365],{"className":5187},[388,389],[344,5189],{"className":5190,"style":488},[487],[344,5192,818],{"className":5193},[492],[344,5195],{"className":5196,"style":488},[487],[344,5198,5200,5204,5278,5281,5321,5361],{"className":5199},[379],[344,5201],{"className":5202,"style":5203},[383],"height:3.106em;vertical-align:-1.2777em;",[344,5205,5209],{"className":5206},[5207,5208],"mop","op-limits",[344,5210,5212,5269],{"className":5211},[517,900],[344,5213,5215,5266],{"className":5214},[521],[344,5216,5219,5241,5254],{"className":5217,"style":5218},[525],"height:1.8283em;",[344,5220,5222,5226],{"style":5221},"top:-1.8723em;margin-left:0em;",[344,5223],{"className":5224,"style":5225},[532],"height:3.05em;",[344,5227,5229],{"className":5228},[537,538,539,540],[344,5230,5232,5235,5238],{"className":5231},[388,540],[344,5233,2234],{"className":5234},[388,389,540],[344,5236,818],{"className":5237},[492,540],[344,5239,828],{"className":5240},[388,540],[344,5242,5244,5247],{"style":5243},"top:-3.05em;",[344,5245],{"className":5246,"style":5225},[532],[344,5248,5249],{},[344,5250,5147],{"className":5251},[5207,5252,5253],"op-symbol","large-op",[344,5255,5257,5260],{"style":5256},"top:-4.3em;margin-left:0em;",[344,5258],{"className":5259,"style":5225},[532],[344,5261,5263],{"className":5262},[537,538,539,540],[344,5264,366],{"className":5265,"style":390},[388,389,540],[344,5267,924],{"className":5268},[923],[344,5270,5272],{"className":5271},[521],[344,5273,5276],{"className":5274,"style":5275},[525],"height:1.2777em;",[344,5277],{},[344,5279],{"className":5280,"style":941},[487],[344,5282,5284,5287],{"className":5283},[388],[344,5285,5162],{"className":5286},[388,389],[344,5288,5290],{"className":5289},[513],[344,5291,5293,5313],{"className":5292},[517,900],[344,5294,5296,5310],{"className":5295},[521],[344,5297,5299],{"className":5298,"style":2471},[525],[344,5300,5301,5304],{"style":910},[344,5302],{"className":5303,"style":533},[532],[344,5305,5307],{"className":5306},[537,538,539,540],[344,5308,2234],{"className":5309},[388,389,540],[344,5311,924],{"className":5312},[923],[344,5314,5316],{"className":5315},[521],[344,5317,5319],{"className":5318,"style":931},[525],[344,5320],{},[344,5322,5324,5327],{"className":5323},[388],[344,5325,4852],{"className":5326,"style":4886},[388,389],[344,5328,5330],{"className":5329},[513],[344,5331,5333,5353],{"className":5332},[517,900],[344,5334,5336,5350],{"className":5335},[521],[344,5337,5339],{"className":5338,"style":2471},[525],[344,5340,5341,5344],{"style":4901},[344,5342],{"className":5343,"style":533},[532],[344,5345,5347],{"className":5346},[537,538,539,540],[344,5348,2234],{"className":5349},[388,389,540],[344,5351,924],{"className":5352},[923],[344,5354,5356],{"className":5355},[521],[344,5357,5359],{"className":5358,"style":931},[525],[344,5360],{},[344,5362,831],{"className":5363},[937],[15,5365,5366],{},"则",[344,5368,5370],{"className":5369,"translate":348},[1047],[344,5371,5373,5456],{"className":5372,"translate":348},[347],[344,5374,5376],{"className":5375},[352],[354,5377,5378],{"xmlns":356,"display":1056},[358,5379,5380,5453],{},[1082,5381,5382],{"width":3486},[1088,5383,5384,5386,5446,5448],{},[1091,5385],{"width":3491},[1091,5387,5388],{},[361,5389,5390,5396,5402,5404,5410,5412,5414,5416,5430,5436,5444],{},[454,5391,5392,5394],{},[364,5393,3365],{},[364,5395,3547],{},[822,5397,5398,5400],{},[364,5399,2175],{},[364,5401,448],{},[450,5403,593],{"stretchy":592},[822,5405,5406,5408],{},[364,5407,815],{},[364,5409,1721],{},[450,5411,602],{"stretchy":592},[364,5413,3365],{},[450,5415,818],{},[5143,5417,5418,5420,5428],{},[450,5419,5147],{},[361,5421,5422,5424,5426],{},[364,5423,2234],{},[450,5425,818],{},[587,5427,828],{},[364,5429,366],{},[822,5431,5432,5434],{},[364,5433,4997],{},[364,5435,2234],{},[2205,5437,5438,5440,5442],{},[364,5439,5162],{},[364,5441,2234],{},[587,5443,589],{},[364,5445,1192],{"mathvariant":1063},[1091,5447],{"width":3491},[1091,5449,5450],{},[3609,5451,5452],{},"(2)",[368,5454,5455],{"encoding":370},"h^T H_f(x_c) h = \\sum_{i=1}^N \\lambda_i a_i^2. \\tag{2}",[344,5457,5459,5593,5762],{"className":5458,"ariaHidden":375},[374],[344,5460,5462,5466,5495,5535,5538,5578,5581,5584,5587,5590],{"className":5461},[379],[344,5463],{"className":5464,"style":5465},[383],"height:1.1774em;vertical-align:-0.2861em;",[344,5467,5469,5472],{"className":5468},[388],[344,5470,3365],{"className":5471},[388,389],[344,5473,5475],{"className":5474},[513],[344,5476,5478],{"className":5477},[517],[344,5479,5481],{"className":5480},[521],[344,5482,5484],{"className":5483,"style":1661},[525],[344,5485,5486,5489],{"style":1682},[344,5487],{"className":5488,"style":533},[532],[344,5490,5492],{"className":5491},[537,538,539,540],[344,5493,3547],{"className":5494,"style":3849},[388,389,540],[344,5496,5498,5501],{"className":5497},[388],[344,5499,2175],{"className":5500,"style":2288},[388,389],[344,5502,5504],{"className":5503},[513],[344,5505,5507,5527],{"className":5506},[517,900],[344,5508,5510,5524],{"className":5509},[521],[344,5511,5513],{"className":5512,"style":2301},[525],[344,5514,5515,5518],{"style":2304},[344,5516],{"className":5517,"style":533},[532],[344,5519,5521],{"className":5520},[537,538,539,540],[344,5522,448],{"className":5523,"style":483},[388,389,540],[344,5525,924],{"className":5526},[923],[344,5528,5530],{"className":5529},[521],[344,5531,5533],{"className":5532,"style":2323},[525],[344,5534],{},[344,5536,593],{"className":5537},[668],[344,5539,5541,5544],{"className":5540},[388],[344,5542,815],{"className":5543},[388,389],[344,5545,5547],{"className":5546},[513],[344,5548,5550,5570],{"className":5549},[517,900],[344,5551,5553,5567],{"className":5552},[521],[344,5554,5556],{"className":5555,"style":1761},[525],[344,5557,5558,5561],{"style":910},[344,5559],{"className":5560,"style":533},[532],[344,5562,5564],{"className":5563},[537,538,539,540],[344,5565,1721],{"className":5566},[388,389,540],[344,5568,924],{"className":5569},[923],[344,5571,5573],{"className":5572},[521],[344,5574,5576],{"className":5575,"style":931},[525],[344,5577],{},[344,5579,602],{"className":5580},[701],[344,5582,3365],{"className":5583},[388,389],[344,5585],{"className":5586,"style":488},[487],[344,5588,818],{"className":5589},[492],[344,5591],{"className":5592,"style":488},[487],[344,5594,5596,5599,5663,5666,5706,5759],{"className":5595},[379],[344,5597],{"className":5598,"style":5203},[383],[344,5600,5602],{"className":5601},[5207,5208],[344,5603,5605,5655],{"className":5604},[517,900],[344,5606,5608,5652],{"className":5607},[521],[344,5609,5611,5631,5641],{"className":5610,"style":5218},[525],[344,5612,5613,5616],{"style":5221},[344,5614],{"className":5615,"style":5225},[532],[344,5617,5619],{"className":5618},[537,538,539,540],[344,5620,5622,5625,5628],{"className":5621},[388,540],[344,5623,2234],{"className":5624},[388,389,540],[344,5626,818],{"className":5627},[492,540],[344,5629,828],{"className":5630},[388,540],[344,5632,5633,5636],{"style":5243},[344,5634],{"className":5635,"style":5225},[532],[344,5637,5638],{},[344,5639,5147],{"className":5640},[5207,5252,5253],[344,5642,5643,5646],{"style":5256},[344,5644],{"className":5645,"style":5225},[532],[344,5647,5649],{"className":5648},[537,538,539,540],[344,5650,366],{"className":5651,"style":390},[388,389,540],[344,5653,924],{"className":5654},[923],[344,5656,5658],{"className":5657},[521],[344,5659,5661],{"className":5660,"style":5275},[525],[344,5662],{},[344,5664],{"className":5665,"style":941},[487],[344,5667,5669,5672],{"className":5668},[388],[344,5670,4997],{"className":5671},[388,389],[344,5673,5675],{"className":5674},[513],[344,5676,5678,5698],{"className":5677},[517,900],[344,5679,5681,5695],{"className":5680},[521],[344,5682,5684],{"className":5683,"style":2471},[525],[344,5685,5686,5689],{"style":910},[344,5687],{"className":5688,"style":533},[532],[344,5690,5692],{"className":5691},[537,538,539,540],[344,5693,2234],{"className":5694},[388,389,540],[344,5696,924],{"className":5697},[923],[344,5699,5701],{"className":5700},[521],[344,5702,5704],{"className":5703,"style":931},[525],[344,5705],{},[344,5707,5709,5712],{"className":5708},[388],[344,5710,5162],{"className":5711},[388,389],[344,5713,5715],{"className":5714},[513],[344,5716,5718,5750],{"className":5717},[517,900],[344,5719,5721,5747],{"className":5720},[521],[344,5722,5724,5736],{"className":5723,"style":2369},[525],[344,5725,5727,5730],{"style":5726},"top:-2.453em;margin-left:0em;margin-right:0.05em;",[344,5728],{"className":5729,"style":533},[532],[344,5731,5733],{"className":5732},[537,538,539,540],[344,5734,2234],{"className":5735},[388,389,540],[344,5737,5738,5741],{"style":1682},[344,5739],{"className":5740,"style":533},[532],[344,5742,5744],{"className":5743},[537,538,539,540],[344,5745,589],{"className":5746},[388,540],[344,5748,924],{"className":5749},[923],[344,5751,5753],{"className":5752},[521],[344,5754,5757],{"className":5755,"style":5756},[525],"height:0.247em;",[344,5758],{},[344,5760,1192],{"className":5761},[388],[344,5763,5765,5768],{"className":5764},[4116],[344,5766],{"className":5767,"style":5203},[383],[344,5769,5771,5774,5780],{"className":5770},[388,4123],[344,5772,593],{"className":5773},[388],[344,5775,5777],{"className":5776},[388],[344,5778,589],{"className":5779},[388],[344,5781,602],{"className":5782},[388],[15,5784,5785,5786,5818],{},"因此，",[324,5787,5788,5789,5817],{},"Hessian 的特征值就是 ",[344,5790,5792,5805],{"className":5791,"translate":348},[347],[344,5793,5795],{"className":5794},[352],[354,5796,5797],{"xmlns":356},[358,5798,5799,5803],{},[361,5800,5801],{},[364,5802,448],{},[368,5804,448],{"encoding":370},[344,5806,5808],{"className":5807,"ariaHidden":375},[374],[344,5809,5811,5814],{"className":5810},[379],[344,5812],{"className":5813,"style":479},[383],[344,5815,448],{"className":5816,"style":483},[388,389]," 在临界点处的主二阶曲率","——每个特征值的符号，直接告诉我们函数沿对应方向「向上弯」还是「向下弯」。",[764,5820,5821,6023,6621,6823,7053],{},[15,5822,5823,5826,5827,5896,5897,6022],{},[324,5824,5825],{},"定义 2.4（非退化极小值、极大值与鞍点）","。设 ",[344,5828,5830,5847],{"className":5829,"translate":348},[347],[344,5831,5833],{"className":5832},[352],[354,5834,5835],{"xmlns":356},[358,5836,5837,5845],{},[361,5838,5839],{},[822,5840,5841,5843],{},[364,5842,815],{},[364,5844,1721],{},[368,5846,4159],{"encoding":370},[344,5848,5850],{"className":5849,"ariaHidden":375},[374],[344,5851,5853,5856],{"className":5852},[379],[344,5854],{"className":5855,"style":3429},[383],[344,5857,5859,5862],{"className":5858},[388],[344,5860,815],{"className":5861},[388,389],[344,5863,5865],{"className":5864},[513],[344,5866,5868,5888],{"className":5867},[517,900],[344,5869,5871,5885],{"className":5870},[521],[344,5872,5874],{"className":5873,"style":1761},[525],[344,5875,5876,5879],{"style":910},[344,5877],{"className":5878,"style":533},[532],[344,5880,5882],{"className":5881},[537,538,539,540],[344,5883,1721],{"className":5884},[388,389,540],[344,5886,924],{"className":5887},[923],[344,5889,5891],{"className":5890},[521],[344,5892,5894],{"className":5893,"style":931},[525],[344,5895],{}," 是非退化临界点，即 ",[344,5898,5900,5927],{"className":5899,"translate":348},[347],[344,5901,5903],{"className":5902},[352],[354,5904,5905],{"xmlns":356},[358,5906,5907,5925],{},[361,5908,5909,5915,5917,5923],{},[822,5910,5911,5913],{},[364,5912,2175],{},[364,5914,448],{},[450,5916,593],{"stretchy":592},[822,5918,5919,5921],{},[364,5920,815],{},[364,5922,1721],{},[450,5924,602],{"stretchy":592},[368,5926,4739],{"encoding":370},[344,5928,5930],{"className":5929,"ariaHidden":375},[374],[344,5931,5933,5936,5976,5979,6019],{"className":5932},[379],[344,5934],{"className":5935,"style":2281},[383],[344,5937,5939,5942],{"className":5938},[388],[344,5940,2175],{"className":5941,"style":2288},[388,389],[344,5943,5945],{"className":5944},[513],[344,5946,5948,5968],{"className":5947},[517,900],[344,5949,5951,5965],{"className":5950},[521],[344,5952,5954],{"className":5953,"style":2301},[525],[344,5955,5956,5959],{"style":2304},[344,5957],{"className":5958,"style":533},[532],[344,5960,5962],{"className":5961},[537,538,539,540],[344,5963,448],{"className":5964,"style":483},[388,389,540],[344,5966,924],{"className":5967},[923],[344,5969,5971],{"className":5970},[521],[344,5972,5974],{"className":5973,"style":2323},[525],[344,5975],{},[344,5977,593],{"className":5978},[668],[344,5980,5982,5985],{"className":5981},[388],[344,5983,815],{"className":5984},[388,389],[344,5986,5988],{"className":5987},[513],[344,5989,5991,6011],{"className":5990},[517,900],[344,5992,5994,6008],{"className":5993},[521],[344,5995,5997],{"className":5996,"style":1761},[525],[344,5998,5999,6002],{"style":910},[344,6000],{"className":6001,"style":533},[532],[344,6003,6005],{"className":6004},[537,538,539,540],[344,6006,1721],{"className":6007},[388,389,540],[344,6009,924],{"className":6010},[923],[344,6012,6014],{"className":6013},[521],[344,6015,6017],{"className":6016,"style":931},[525],[344,6018],{},[344,6020,602],{"className":6021},[701]," 没有零特征值。",[6024,6025,6026,6225,6423],"ul",{},[317,6027,6028,6029,6154,6155,6224],{},"若 ",[344,6030,6032,6059],{"className":6031,"translate":348},[347],[344,6033,6035],{"className":6034},[352],[354,6036,6037],{"xmlns":356},[358,6038,6039,6057],{},[361,6040,6041,6047,6049,6055],{},[822,6042,6043,6045],{},[364,6044,2175],{},[364,6046,448],{},[450,6048,593],{"stretchy":592},[822,6050,6051,6053],{},[364,6052,815],{},[364,6054,1721],{},[450,6056,602],{"stretchy":592},[368,6058,4739],{"encoding":370},[344,6060,6062],{"className":6061,"ariaHidden":375},[374],[344,6063,6065,6068,6108,6111,6151],{"className":6064},[379],[344,6066],{"className":6067,"style":2281},[383],[344,6069,6071,6074],{"className":6070},[388],[344,6072,2175],{"className":6073,"style":2288},[388,389],[344,6075,6077],{"className":6076},[513],[344,6078,6080,6100],{"className":6079},[517,900],[344,6081,6083,6097],{"className":6082},[521],[344,6084,6086],{"className":6085,"style":2301},[525],[344,6087,6088,6091],{"style":2304},[344,6089],{"className":6090,"style":533},[532],[344,6092,6094],{"className":6093},[537,538,539,540],[344,6095,448],{"className":6096,"style":483},[388,389,540],[344,6098,924],{"className":6099},[923],[344,6101,6103],{"className":6102},[521],[344,6104,6106],{"className":6105,"style":2323},[525],[344,6107],{},[344,6109,593],{"className":6110},[668],[344,6112,6114,6117],{"className":6113},[388],[344,6115,815],{"className":6116},[388,389],[344,6118,6120],{"className":6119},[513],[344,6121,6123,6143],{"className":6122},[517,900],[344,6124,6126,6140],{"className":6125},[521],[344,6127,6129],{"className":6128,"style":1761},[525],[344,6130,6131,6134],{"style":910},[344,6132],{"className":6133,"style":533},[532],[344,6135,6137],{"className":6136},[537,538,539,540],[344,6138,1721],{"className":6139},[388,389,540],[344,6141,924],{"className":6142},[923],[344,6144,6146],{"className":6145},[521],[344,6147,6149],{"className":6148,"style":931},[525],[344,6150],{},[344,6152,602],{"className":6153},[701]," 的所有特征值都为正，则 ",[344,6156,6158,6175],{"className":6157,"translate":348},[347],[344,6159,6161],{"className":6160},[352],[354,6162,6163],{"xmlns":356},[358,6164,6165,6173],{},[361,6166,6167],{},[822,6168,6169,6171],{},[364,6170,815],{},[364,6172,1721],{},[368,6174,4159],{"encoding":370},[344,6176,6178],{"className":6177,"ariaHidden":375},[374],[344,6179,6181,6184],{"className":6180},[379],[344,6182],{"className":6183,"style":3429},[383],[344,6185,6187,6190],{"className":6186},[388],[344,6188,815],{"className":6189},[388,389],[344,6191,6193],{"className":6192},[513],[344,6194,6196,6216],{"className":6195},[517,900],[344,6197,6199,6213],{"className":6198},[521],[344,6200,6202],{"className":6201,"style":1761},[525],[344,6203,6204,6207],{"style":910},[344,6205],{"className":6206,"style":533},[532],[344,6208,6210],{"className":6209},[537,538,539,540],[344,6211,1721],{"className":6212},[388,389,540],[344,6214,924],{"className":6215},[923],[344,6217,6219],{"className":6218},[521],[344,6220,6222],{"className":6221,"style":931},[525],[344,6223],{}," 是非退化局部极小值。",[317,6226,6028,6227,6352,6353,6422],{},[344,6228,6230,6257],{"className":6229,"translate":348},[347],[344,6231,6233],{"className":6232},[352],[354,6234,6235],{"xmlns":356},[358,6236,6237,6255],{},[361,6238,6239,6245,6247,6253],{},[822,6240,6241,6243],{},[364,6242,2175],{},[364,6244,448],{},[450,6246,593],{"stretchy":592},[822,6248,6249,6251],{},[364,6250,815],{},[364,6252,1721],{},[450,6254,602],{"stretchy":592},[368,6256,4739],{"encoding":370},[344,6258,6260],{"className":6259,"ariaHidden":375},[374],[344,6261,6263,6266,6306,6309,6349],{"className":6262},[379],[344,6264],{"className":6265,"style":2281},[383],[344,6267,6269,6272],{"className":6268},[388],[344,6270,2175],{"className":6271,"style":2288},[388,389],[344,6273,6275],{"className":6274},[513],[344,6276,6278,6298],{"className":6277},[517,900],[344,6279,6281,6295],{"className":6280},[521],[344,6282,6284],{"className":6283,"style":2301},[525],[344,6285,6286,6289],{"style":2304},[344,6287],{"className":6288,"style":533},[532],[344,6290,6292],{"className":6291},[537,538,539,540],[344,6293,448],{"className":6294,"style":483},[388,389,540],[344,6296,924],{"className":6297},[923],[344,6299,6301],{"className":6300},[521],[344,6302,6304],{"className":6303,"style":2323},[525],[344,6305],{},[344,6307,593],{"className":6308},[668],[344,6310,6312,6315],{"className":6311},[388],[344,6313,815],{"className":6314},[388,389],[344,6316,6318],{"className":6317},[513],[344,6319,6321,6341],{"className":6320},[517,900],[344,6322,6324,6338],{"className":6323},[521],[344,6325,6327],{"className":6326,"style":1761},[525],[344,6328,6329,6332],{"style":910},[344,6330],{"className":6331,"style":533},[532],[344,6333,6335],{"className":6334},[537,538,539,540],[344,6336,1721],{"className":6337},[388,389,540],[344,6339,924],{"className":6340},[923],[344,6342,6344],{"className":6343},[521],[344,6345,6347],{"className":6346,"style":931},[525],[344,6348],{},[344,6350,602],{"className":6351},[701]," 的所有特征值都为负，则 ",[344,6354,6356,6373],{"className":6355,"translate":348},[347],[344,6357,6359],{"className":6358},[352],[354,6360,6361],{"xmlns":356},[358,6362,6363,6371],{},[361,6364,6365],{},[822,6366,6367,6369],{},[364,6368,815],{},[364,6370,1721],{},[368,6372,4159],{"encoding":370},[344,6374,6376],{"className":6375,"ariaHidden":375},[374],[344,6377,6379,6382],{"className":6378},[379],[344,6380],{"className":6381,"style":3429},[383],[344,6383,6385,6388],{"className":6384},[388],[344,6386,815],{"className":6387},[388,389],[344,6389,6391],{"className":6390},[513],[344,6392,6394,6414],{"className":6393},[517,900],[344,6395,6397,6411],{"className":6396},[521],[344,6398,6400],{"className":6399,"style":1761},[525],[344,6401,6402,6405],{"style":910},[344,6403],{"className":6404,"style":533},[532],[344,6406,6408],{"className":6407},[537,538,539,540],[344,6409,1721],{"className":6410},[388,389,540],[344,6412,924],{"className":6413},[923],[344,6415,6417],{"className":6416},[521],[344,6418,6420],{"className":6419,"style":931},[525],[344,6421],{}," 是非退化局部极大值。",[317,6424,6028,6425,6550,6551,6620],{},[344,6426,6428,6455],{"className":6427,"translate":348},[347],[344,6429,6431],{"className":6430},[352],[354,6432,6433],{"xmlns":356},[358,6434,6435,6453],{},[361,6436,6437,6443,6445,6451],{},[822,6438,6439,6441],{},[364,6440,2175],{},[364,6442,448],{},[450,6444,593],{"stretchy":592},[822,6446,6447,6449],{},[364,6448,815],{},[364,6450,1721],{},[450,6452,602],{"stretchy":592},[368,6454,4739],{"encoding":370},[344,6456,6458],{"className":6457,"ariaHidden":375},[374],[344,6459,6461,6464,6504,6507,6547],{"className":6460},[379],[344,6462],{"className":6463,"style":2281},[383],[344,6465,6467,6470],{"className":6466},[388],[344,6468,2175],{"className":6469,"style":2288},[388,389],[344,6471,6473],{"className":6472},[513],[344,6474,6476,6496],{"className":6475},[517,900],[344,6477,6479,6493],{"className":6478},[521],[344,6480,6482],{"className":6481,"style":2301},[525],[344,6483,6484,6487],{"style":2304},[344,6485],{"className":6486,"style":533},[532],[344,6488,6490],{"className":6489},[537,538,539,540],[344,6491,448],{"className":6492,"style":483},[388,389,540],[344,6494,924],{"className":6495},[923],[344,6497,6499],{"className":6498},[521],[344,6500,6502],{"className":6501,"style":2323},[525],[344,6503],{},[344,6505,593],{"className":6506},[668],[344,6508,6510,6513],{"className":6509},[388],[344,6511,815],{"className":6512},[388,389],[344,6514,6516],{"className":6515},[513],[344,6517,6519,6539],{"className":6518},[517,900],[344,6520,6522,6536],{"className":6521},[521],[344,6523,6525],{"className":6524,"style":1761},[525],[344,6526,6527,6530],{"style":910},[344,6528],{"className":6529,"style":533},[532],[344,6531,6533],{"className":6532},[537,538,539,540],[344,6534,1721],{"className":6535},[388,389,540],[344,6537,924],{"className":6538},[923],[344,6540,6542],{"className":6541},[521],[344,6543,6545],{"className":6544,"style":931},[525],[344,6546],{},[344,6548,602],{"className":6549},[701]," 至少有一个正特征值和至少一个负特征值，则 ",[344,6552,6554,6571],{"className":6553,"translate":348},[347],[344,6555,6557],{"className":6556},[352],[354,6558,6559],{"xmlns":356},[358,6560,6561,6569],{},[361,6562,6563],{},[822,6564,6565,6567],{},[364,6566,815],{},[364,6568,1721],{},[368,6570,4159],{"encoding":370},[344,6572,6574],{"className":6573,"ariaHidden":375},[374],[344,6575,6577,6580],{"className":6576},[379],[344,6578],{"className":6579,"style":3429},[383],[344,6581,6583,6586],{"className":6582},[388],[344,6584,815],{"className":6585},[388,389],[344,6587,6589],{"className":6588},[513],[344,6590,6592,6612],{"className":6591},[517,900],[344,6593,6595,6609],{"className":6594},[521],[344,6596,6598],{"className":6597,"style":1761},[525],[344,6599,6600,6603],{"style":910},[344,6601],{"className":6602,"style":533},[532],[344,6604,6606],{"className":6605},[537,538,539,540],[344,6607,1721],{"className":6608},[388,389,540],[344,6610,924],{"className":6611},[923],[344,6613,6615],{"className":6614},[521],[344,6616,6618],{"className":6617,"style":931},[525],[344,6619],{}," 是鞍点。",[15,6622,6623,6626,6627,6696,6697,6822],{},[324,6624,6625],{},"定义 2.5（Morse 指数）","。非退化临界点 ",[344,6628,6630,6647],{"className":6629,"translate":348},[347],[344,6631,6633],{"className":6632},[352],[354,6634,6635],{"xmlns":356},[358,6636,6637,6645],{},[361,6638,6639],{},[822,6640,6641,6643],{},[364,6642,815],{},[364,6644,1721],{},[368,6646,4159],{"encoding":370},[344,6648,6650],{"className":6649,"ariaHidden":375},[374],[344,6651,6653,6656],{"className":6652},[379],[344,6654],{"className":6655,"style":3429},[383],[344,6657,6659,6662],{"className":6658},[388],[344,6660,815],{"className":6661},[388,389],[344,6663,6665],{"className":6664},[513],[344,6666,6668,6688],{"className":6667},[517,900],[344,6669,6671,6685],{"className":6670},[521],[344,6672,6674],{"className":6673,"style":1761},[525],[344,6675,6676,6679],{"style":910},[344,6677],{"className":6678,"style":533},[532],[344,6680,6682],{"className":6681},[537,538,539,540],[344,6683,1721],{"className":6684},[388,389,540],[344,6686,924],{"className":6687},[923],[344,6689,6691],{"className":6690},[521],[344,6692,6694],{"className":6693,"style":931},[525],[344,6695],{}," 的 Morse 指数是 ",[344,6698,6700,6727],{"className":6699,"translate":348},[347],[344,6701,6703],{"className":6702},[352],[354,6704,6705],{"xmlns":356},[358,6706,6707,6725],{},[361,6708,6709,6715,6717,6723],{},[822,6710,6711,6713],{},[364,6712,2175],{},[364,6714,448],{},[450,6716,593],{"stretchy":592},[822,6718,6719,6721],{},[364,6720,815],{},[364,6722,1721],{},[450,6724,602],{"stretchy":592},[368,6726,4739],{"encoding":370},[344,6728,6730],{"className":6729,"ariaHidden":375},[374],[344,6731,6733,6736,6776,6779,6819],{"className":6732},[379],[344,6734],{"className":6735,"style":2281},[383],[344,6737,6739,6742],{"className":6738},[388],[344,6740,2175],{"className":6741,"style":2288},[388,389],[344,6743,6745],{"className":6744},[513],[344,6746,6748,6768],{"className":6747},[517,900],[344,6749,6751,6765],{"className":6750},[521],[344,6752,6754],{"className":6753,"style":2301},[525],[344,6755,6756,6759],{"style":2304},[344,6757],{"className":6758,"style":533},[532],[344,6760,6762],{"className":6761},[537,538,539,540],[344,6763,448],{"className":6764,"style":483},[388,389,540],[344,6766,924],{"className":6767},[923],[344,6769,6771],{"className":6770},[521],[344,6772,6774],{"className":6773,"style":2323},[525],[344,6775],{},[344,6777,593],{"className":6778},[668],[344,6780,6782,6785],{"className":6781},[388],[344,6783,815],{"className":6784},[388,389],[344,6786,6788],{"className":6787},[513],[344,6789,6791,6811],{"className":6790},[517,900],[344,6792,6794,6808],{"className":6793},[521],[344,6795,6797],{"className":6796,"style":1761},[525],[344,6798,6799,6802],{"style":910},[344,6800],{"className":6801,"style":533},[532],[344,6803,6805],{"className":6804},[537,538,539,540],[344,6806,1721],{"className":6807},[388,389,540],[344,6809,924],{"className":6810},[923],[344,6812,6814],{"className":6813},[521],[344,6815,6817],{"className":6816,"style":931},[525],[344,6818],{},[344,6820,602],{"className":6821},[701]," 的负特征值个数",[344,6824,6826],{"className":6825,"translate":348},[1047],[344,6827,6829,6887],{"className":6828,"translate":348},[347],[344,6830,6832],{"className":6831},[352],[354,6833,6834],{"xmlns":356,"display":1056},[358,6835,6836,6884],{},[361,6837,6838,6841,6844,6846,6852,6854,6856,6858,6861,6864,6866,6868,6874,6877,6879,6882],{},[364,6839,6840],{"mathvariant":1063},"index",[450,6842,6843],{},"⁡",[450,6845,593],{"stretchy":592},[822,6847,6848,6850],{},[364,6849,815],{},[364,6851,1721],{},[450,6853,602],{"stretchy":592},[450,6855,452],{},[450,6857,818],{},[364,6859,6860],{"mathvariant":1063},"#",[450,6862,6863],{"stretchy":592},"{",[364,6865,2234],{},[450,6867,452],{},[822,6869,6870,6872],{},[364,6871,4997],{},[364,6873,2234],{},[450,6875,6876],{},"\u003C",[587,6878,1096],{},[450,6880,6881],{"stretchy":592},"}",[364,6883,1192],{"mathvariant":1063},[368,6885,6886],{"encoding":370},"\\operatorname{index}(x_c) := \\#\\{i : \\lambda_i \u003C 0\\}.",[344,6888,6890,6958,6982,7038],{"className":6889,"ariaHidden":375},[374],[344,6891,6893,6896,6903,6906,6946,6949,6952,6955],{"className":6892},[379],[344,6894],{"className":6895,"style":884},[383],[344,6897,6899],{"className":6898},[5207],[344,6900,6840],{"className":6901},[388,6902],"mathrm",[344,6904,593],{"className":6905},[668],[344,6907,6909,6912],{"className":6908},[388],[344,6910,815],{"className":6911},[388,389],[344,6913,6915],{"className":6914},[513],[344,6916,6918,6938],{"className":6917},[517,900],[344,6919,6921,6935],{"className":6920},[521],[344,6922,6924],{"className":6923,"style":1761},[525],[344,6925,6926,6929],{"style":910},[344,6927],{"className":6928,"style":533},[532],[344,6930,6932],{"className":6931},[537,538,539,540],[344,6933,1721],{"className":6934},[388,389,540],[344,6936,924],{"className":6937},[923],[344,6939,6941],{"className":6940},[521],[344,6942,6944],{"className":6943,"style":931},[525],[344,6945],{},[344,6947,602],{"className":6948},[701],[344,6950],{"className":6951,"style":488},[487],[344,6953,1226],{"className":6954},[492],[344,6956],{"className":6957,"style":488},[487],[344,6959,6961,6964,6967,6970,6973,6976,6979],{"className":6960},[379],[344,6962],{"className":6963,"style":884},[383],[344,6965,6860],{"className":6966},[388],[344,6968,6863],{"className":6969},[668],[344,6971,2234],{"className":6972},[388,389],[344,6974],{"className":6975,"style":488},[487],[344,6977,452],{"className":6978},[492],[344,6980],{"className":6981,"style":488},[487],[344,6983,6985,6989,7029,7032,7035],{"className":6984},[379],[344,6986],{"className":6987,"style":6988},[383],"height:0.8444em;vertical-align:-0.15em;",[344,6990,6992,6995],{"className":6991},[388],[344,6993,4997],{"className":6994},[388,389],[344,6996,6998],{"className":6997},[513],[344,6999,7001,7021],{"className":7000},[517,900],[344,7002,7004,7018],{"className":7003},[521],[344,7005,7007],{"className":7006,"style":2471},[525],[344,7008,7009,7012],{"style":910},[344,7010],{"className":7011,"style":533},[532],[344,7013,7015],{"className":7014},[537,538,539,540],[344,7016,2234],{"className":7017},[388,389,540],[344,7019,924],{"className":7020},[923],[344,7022,7024],{"className":7023},[521],[344,7025,7027],{"className":7026,"style":931},[525],[344,7028],{},[344,7030],{"className":7031,"style":488},[487],[344,7033,6876],{"className":7034},[492],[344,7036],{"className":7037,"style":488},[487],[344,7039,7041,7044,7047,7050],{"className":7040},[379],[344,7042],{"className":7043,"style":884},[383],[344,7045,1096],{"className":7046},[388],[344,7048,6881],{"className":7049},[701],[344,7051,1192],{"className":7052},[388],[15,7054,7055,7056,7084,7085,7137],{},"非退化局部极小值的指数为 0，非退化局部极大值的指数为 ",[344,7057,7059,7072],{"className":7058,"translate":348},[347],[344,7060,7062],{"className":7061},[352],[354,7063,7064],{"xmlns":356},[358,7065,7066,7070],{},[361,7067,7068],{},[364,7069,366],{},[368,7071,366],{"encoding":370},[344,7073,7075],{"className":7074,"ariaHidden":375},[374],[344,7076,7078,7081],{"className":7077},[379],[344,7079],{"className":7080,"style":384},[383],[344,7082,366],{"className":7083,"style":390},[388,389],"，鞍点的指数在 1 到 ",[344,7086,7088,7106],{"className":7087,"translate":348},[347],[344,7089,7091],{"className":7090},[352],[354,7092,7093],{"xmlns":356},[358,7094,7095,7103],{},[361,7096,7097,7099,7101],{},[364,7098,366],{},[450,7100,3372],{},[587,7102,828],{},[368,7104,7105],{"encoding":370},"N-1",[344,7107,7109,7127],{"className":7108,"ariaHidden":375},[374],[344,7110,7112,7115,7118,7121,7124],{"className":7111},[379],[344,7113],{"className":7114,"style":2131},[383],[344,7116,366],{"className":7117,"style":390},[388,389],[344,7119],{"className":7120,"style":2138},[487],[344,7122,3372],{"className":7123},[2142],[344,7125],{"className":7126,"style":2138},[487],[344,7128,7130,7134],{"className":7129},[379],[344,7131],{"className":7132,"style":7133},[383],"height:0.6444em;",[344,7135,828],{"className":7136},[388]," 之间。",[15,7139,7140,7141,7169,7170,7198],{},"高维的困难此刻清晰可见。在 1 维中，非退化极小值只要求一个正的二阶导数。在 ",[344,7142,7144,7157],{"className":7143,"translate":348},[347],[344,7145,7147],{"className":7146},[352],[354,7148,7149],{"xmlns":356},[358,7150,7151,7155],{},[361,7152,7153],{},[364,7154,366],{},[368,7156,366],{"encoding":370},[344,7158,7160],{"className":7159,"ariaHidden":375},[374],[344,7161,7163,7166],{"className":7162},[379],[344,7164],{"className":7165,"style":384},[383],[344,7167,366],{"className":7168,"style":390},[388,389]," 维中，非退化局部极小值要求 ",[344,7171,7173,7186],{"className":7172,"translate":348},[347],[344,7174,7176],{"className":7175},[352],[354,7177,7178],{"xmlns":356},[358,7179,7180,7184],{},[361,7181,7182],{},[364,7183,366],{},[368,7185,366],{"encoding":370},[344,7187,7189],{"className":7188,"ariaHidden":375},[374],[344,7190,7192,7195],{"className":7191},[379],[344,7193],{"className":7194,"style":384},[383],[344,7196,366],{"className":7197,"style":390},[388,389]," 个同时成立的正曲率条件：",[344,7200,7202],{"className":7201,"translate":348},[1047],[344,7203,7205,7264],{"className":7204,"translate":348},[347],[344,7206,7208],{"className":7207},[352],[354,7209,7210],{"xmlns":356,"display":1056},[358,7211,7212,7261],{},[361,7213,7214,7220,7223,7225,7227,7230,7236,7238,7240,7242,7244,7246,7248,7250,7256,7258],{},[822,7215,7216,7218],{},[364,7217,4997],{},[587,7219,828],{},[450,7221,7222],{},">",[587,7224,1096],{},[450,7226,831],{"separator":375},[3609,7228,7229],{}," ",[822,7231,7232,7234],{},[364,7233,4997],{},[587,7235,589],{},[450,7237,7222],{},[587,7239,1096],{},[450,7241,831],{"separator":375},[3609,7243,7229],{},[450,7245,834],{},[450,7247,831],{"separator":375},[3609,7249,7229],{},[822,7251,7252,7254],{},[364,7253,4997],{},[364,7255,366],{},[450,7257,7222],{},[587,7259,7260],{},"0.",[368,7262,7263],{"encoding":370},"\\lambda_1 > 0,\\ \\lambda_2 > 0,\\ \\dots,\\ \\lambda_N > 0.",[344,7265,7267,7322,7389,7471],{"className":7266,"ariaHidden":375},[374],[344,7268,7270,7273,7313,7316,7319],{"className":7269},[379],[344,7271],{"className":7272,"style":6988},[383],[344,7274,7276,7279],{"className":7275},[388],[344,7277,4997],{"className":7278},[388,389],[344,7280,7282],{"className":7281},[513],[344,7283,7285,7305],{"className":7284},[517,900],[344,7286,7288,7302],{"className":7287},[521],[344,7289,7291],{"className":7290,"style":907},[525],[344,7292,7293,7296],{"style":910},[344,7294],{"className":7295,"style":533},[532],[344,7297,7299],{"className":7298},[537,538,539,540],[344,7300,828],{"className":7301},[388,540],[344,7303,924],{"className":7304},[923],[344,7306,7308],{"className":7307},[521],[344,7309,7311],{"className":7310,"style":931},[525],[344,7312],{},[344,7314],{"className":7315,"style":488},[487],[344,7317,7222],{"className":7318},[492],[344,7320],{"className":7321,"style":488},[487],[344,7323,7325,7328,7331,7334,7337,7340,7380,7383,7386],{"className":7324},[379],[344,7326],{"className":7327,"style":479},[383],[344,7329,1096],{"className":7330},[388],[344,7332,831],{"className":7333},[937],[344,7335,7229],{"className":7336},[487],[344,7338],{"className":7339,"style":941},[487],[344,7341,7343,7346],{"className":7342},[388],[344,7344,4997],{"className":7345},[388,389],[344,7347,7349],{"className":7348},[513],[344,7350,7352,7372],{"className":7351},[517,900],[344,7353,7355,7369],{"className":7354},[521],[344,7356,7358],{"className":7357,"style":907},[525],[344,7359,7360,7363],{"style":910},[344,7361],{"className":7362,"style":533},[532],[344,7364,7366],{"className":7365},[537,538,539,540],[344,7367,589],{"className":7368},[388,540],[344,7370,924],{"className":7371},[923],[344,7373,7375],{"className":7374},[521],[344,7376,7378],{"className":7377,"style":931},[525],[344,7379],{},[344,7381],{"className":7382,"style":488},[487],[344,7384,7222],{"className":7385},[492],[344,7387],{"className":7388,"style":488},[487],[344,7390,7392,7395,7398,7401,7404,7407,7410,7413,7416,7419,7422,7462,7465,7468],{"className":7391},[379],[344,7393],{"className":7394,"style":479},[383],[344,7396,1096],{"className":7397},[388],[344,7399,831],{"className":7400},[937],[344,7402,7229],{"className":7403},[487],[344,7405],{"className":7406,"style":941},[487],[344,7408,834],{"className":7409},[945],[344,7411],{"className":7412,"style":941},[487],[344,7414,831],{"className":7415},[937],[344,7417,7229],{"className":7418},[487],[344,7420],{"className":7421,"style":941},[487],[344,7423,7425,7428],{"className":7424},[388],[344,7426,4997],{"className":7427},[388,389],[344,7429,7431],{"className":7430},[513],[344,7432,7434,7454],{"className":7433},[517,900],[344,7435,7437,7451],{"className":7436},[521],[344,7438,7440],{"className":7439,"style":973},[525],[344,7441,7442,7445],{"style":910},[344,7443],{"className":7444,"style":533},[532],[344,7446,7448],{"className":7447},[537,538,539,540],[344,7449,366],{"className":7450,"style":390},[388,389,540],[344,7452,924],{"className":7453},[923],[344,7455,7457],{"className":7456},[521],[344,7458,7460],{"className":7459,"style":931},[525],[344,7461],{},[344,7463],{"className":7464,"style":488},[487],[344,7466,7222],{"className":7467},[492],[344,7469],{"className":7470,"style":488},[487],[344,7472,7474,7477],{"className":7473},[379],[344,7475],{"className":7476,"style":7133},[383],[344,7478,7260],{"className":7479},[388],[15,7481,7482],{},"相比之下，鞍点只要求至少一个正特征值和至少一个负特征值。当 Hessian 是随机的、且对正曲率没有强偏好时，后者显然容易满足得多。",[11,7484,7485],{"id":7485},"临界点本身就稀少",[15,7487,7488],{},"在讨论「临界点是不是极小值」之前，先确认一个更基础、但常被忽视的事实：临界点本身在空间中就是「测度为零」的稀疏集合。这个结论不难证明，但它还不是本文要讲的高维现象——它对一维函数同样成立。",[764,7490,7491,7729,7937,7969],{},[15,7492,7493,1702,7496,7554,7555,7658,7659,7728],{},[324,7494,7495],{},"定义 3.1（Morse 函数）",[344,7497,7499,7516],{"className":7498,"translate":348},[347],[344,7500,7502],{"className":7501},[352],[354,7503,7504],{"xmlns":356},[358,7505,7506,7514],{},[361,7507,7508],{},[454,7509,7510,7512],{},[364,7511,585],{},[587,7513,589],{},[368,7515,723],{"encoding":370},[344,7517,7519],{"className":7518,"ariaHidden":375},[374],[344,7520,7522,7525],{"className":7521},[379],[344,7523],{"className":7524,"style":653},[383],[344,7526,7528,7531],{"className":7527},[388],[344,7529,585],{"className":7530,"style":640},[388,389],[344,7532,7534],{"className":7533},[513],[344,7535,7537],{"className":7536},[517],[344,7538,7540],{"className":7539},[521],[344,7541,7543],{"className":7542,"style":653},[525],[344,7544,7545,7548],{"style":528},[344,7546],{"className":7547,"style":533},[532],[344,7549,7551],{"className":7550},[537,538,539,540],[344,7552,589],{"className":7553},[388,540]," 函数 ",[344,7556,7558,7584],{"className":7557,"translate":348},[347],[344,7559,7561],{"className":7560},[352],[354,7562,7563],{"xmlns":356},[358,7564,7565,7581],{},[361,7566,7567,7569,7571,7577,7579],{},[364,7568,448],{},[450,7570,452],{},[454,7572,7573,7575],{},[364,7574,459],{"mathvariant":458},[364,7576,366],{},[450,7578,464],{},[364,7580,459],{"mathvariant":458},[368,7582,7583],{"encoding":370},"f:\\mathbb{R}^N\\to\\mathbb{R}",[344,7585,7587,7605,7649],{"className":7586,"ariaHidden":375},[374],[344,7588,7590,7593,7596,7599,7602],{"className":7589},[379],[344,7591],{"className":7592,"style":479},[383],[344,7594,448],{"className":7595,"style":483},[388,389],[344,7597],{"className":7598,"style":488},[487],[344,7600,452],{"className":7601},[492],[344,7603],{"className":7604,"style":488},[487],[344,7606,7608,7611,7640,7643,7646],{"className":7607},[379],[344,7609],{"className":7610,"style":502},[383],[344,7612,7614,7617],{"className":7613},[388],[344,7615,459],{"className":7616},[388,509],[344,7618,7620],{"className":7619},[513],[344,7621,7623],{"className":7622},[517],[344,7624,7626],{"className":7625},[521],[344,7627,7629],{"className":7628,"style":502},[525],[344,7630,7631,7634],{"style":528},[344,7632],{"className":7633,"style":533},[532],[344,7635,7637],{"className":7636},[537,538,539,540],[344,7638,366],{"className":7639,"style":390},[388,389,540],[344,7641],{"className":7642,"style":488},[487],[344,7644,464],{"className":7645},[492],[344,7647],{"className":7648,"style":488},[487],[344,7650,7652,7655],{"className":7651},[379],[344,7653],{"className":7654,"style":559},[383],[344,7656,459],{"className":7657},[388,509]," 的每个临界点都是非退化的，即对每个临界点 ",[344,7660,7662,7679],{"className":7661,"translate":348},[347],[344,7663,7665],{"className":7664},[352],[354,7666,7667],{"xmlns":356},[358,7668,7669,7677],{},[361,7670,7671],{},[822,7672,7673,7675],{},[364,7674,815],{},[364,7676,1721],{},[368,7678,4159],{"encoding":370},[344,7680,7682],{"className":7681,"ariaHidden":375},[374],[344,7683,7685,7688],{"className":7684},[379],[344,7686],{"className":7687,"style":3429},[383],[344,7689,7691,7694],{"className":7690},[388],[344,7692,815],{"className":7693},[388,389],[344,7695,7697],{"className":7696},[513],[344,7698,7700,7720],{"className":7699},[517,900],[344,7701,7703,7717],{"className":7702},[521],[344,7704,7706],{"className":7705,"style":1761},[525],[344,7707,7708,7711],{"style":910},[344,7709],{"className":7710,"style":533},[532],[344,7712,7714],{"className":7713},[537,538,539,540],[344,7715,1721],{"className":7716},[388,389,540],[344,7718,924],{"className":7719},[923],[344,7721,7723],{"className":7722},[521],[344,7724,7726],{"className":7725,"style":931},[525],[344,7727],{}," 有",[344,7730,7732],{"className":7731,"translate":348},[1047],[344,7733,7735,7775],{"className":7734,"translate":348},[347],[344,7736,7738],{"className":7737},[352],[354,7739,7740],{"xmlns":356,"display":1056},[358,7741,7742,7772],{},[361,7743,7744,7747,7749,7755,7757,7763,7765,7768,7770],{},[364,7745,7746],{},"det",[450,7748,6843],{},[822,7750,7751,7753],{},[364,7752,2175],{},[364,7754,448],{},[450,7756,593],{"stretchy":592},[822,7758,7759,7761],{},[364,7760,815],{},[364,7762,1721],{},[450,7764,602],{"stretchy":592},[450,7766,7767],{"mathvariant":1063},"≠",[587,7769,1096],{},[450,7771,831],{"separator":375},[368,7773,7774],{"encoding":370},"\\det H_f(x_c) \\neq 0,",[344,7776,7778,7925],{"className":7777,"ariaHidden":375},[374],[344,7779,7781,7784,7787,7790,7830,7833,7873,7876,7879,7922],{"className":7780},[379],[344,7782],{"className":7783,"style":2281},[383],[344,7785,7746],{"className":7786},[5207],[344,7788],{"className":7789,"style":941},[487],[344,7791,7793,7796],{"className":7792},[388],[344,7794,2175],{"className":7795,"style":2288},[388,389],[344,7797,7799],{"className":7798},[513],[344,7800,7802,7822],{"className":7801},[517,900],[344,7803,7805,7819],{"className":7804},[521],[344,7806,7808],{"className":7807,"style":2301},[525],[344,7809,7810,7813],{"style":2304},[344,7811],{"className":7812,"style":533},[532],[344,7814,7816],{"className":7815},[537,538,539,540],[344,7817,448],{"className":7818,"style":483},[388,389,540],[344,7820,924],{"className":7821},[923],[344,7823,7825],{"className":7824},[521],[344,7826,7828],{"className":7827,"style":2323},[525],[344,7829],{},[344,7831,593],{"className":7832},[668],[344,7834,7836,7839],{"className":7835},[388],[344,7837,815],{"className":7838},[388,389],[344,7840,7842],{"className":7841},[513],[344,7843,7845,7865],{"className":7844},[517,900],[344,7846,7848,7862],{"className":7847},[521],[344,7849,7851],{"className":7850,"style":1761},[525],[344,7852,7853,7856],{"style":910},[344,7854],{"className":7855,"style":533},[532],[344,7857,7859],{"className":7858},[537,538,539,540],[344,7860,1721],{"className":7861},[388,389,540],[344,7863,924],{"className":7864},[923],[344,7866,7868],{"className":7867},[521],[344,7869,7871],{"className":7870,"style":931},[525],[344,7872],{},[344,7874,602],{"className":7875},[701],[344,7877],{"className":7878,"style":488},[487],[344,7880,7882,7915,7919],{"className":7881},[492],[344,7883,7885],{"className":7884},[492],[344,7886,7889],{"className":7887},[388,7888],"vbox",[344,7890,7893],{"className":7891},[7892],"thinbox",[344,7894,7897,7900,7911],{"className":7895},[7896],"rlap",[344,7898],{"className":7899,"style":479},[383],[344,7901,7904],{"className":7902},[7903],"inner",[344,7905,7907],{"className":7906},[388],[344,7908,7910],{"className":7909},[492],"",[344,7912],{"className":7913},[7914],"fix",[344,7916],{"className":7917},[487,7918],"nobreak",[344,7920,818],{"className":7921},[492],[344,7923],{"className":7924,"style":488},[487],[344,7926,7928,7931,7934],{"className":7927},[379],[344,7929],{"className":7930,"style":1943},[383],[344,7932,1096],{"className":7933},[388],[344,7935,831],{"className":7936},[937],[15,7938,7939,7940,7968],{},"则称 ",[344,7941,7943,7956],{"className":7942,"translate":348},[347],[344,7944,7946],{"className":7945},[352],[354,7947,7948],{"xmlns":356},[358,7949,7950,7954],{},[361,7951,7952],{},[364,7953,448],{},[368,7955,448],{"encoding":370},[344,7957,7959],{"className":7958,"ariaHidden":375},[374],[344,7960,7962,7965],{"className":7961},[379],[344,7963],{"className":7964,"style":479},[383],[344,7966,448],{"className":7967,"style":483},[388,389]," 为 Morse 函数。",[15,7970,7971,1702,7974,8002,8003,8062],{},[324,7972,7973],{},"命题 3.2",[344,7975,7977,7990],{"className":7976,"translate":348},[347],[344,7978,7980],{"className":7979},[352],[354,7981,7982],{"xmlns":356},[358,7983,7984,7988],{},[361,7985,7986],{},[364,7987,448],{},[368,7989,448],{"encoding":370},[344,7991,7993],{"className":7992,"ariaHidden":375},[374],[344,7994,7996,7999],{"className":7995},[379],[344,7997],{"className":7998,"style":479},[383],[344,8000,448],{"className":8001,"style":483},[388,389]," 是 Morse 函数，则其临界点是孤立的。特别地，在 ",[344,8004,8006,8024],{"className":8005,"translate":348},[347],[344,8007,8009],{"className":8008},[352],[354,8010,8011],{"xmlns":356},[358,8012,8013,8021],{},[361,8014,8015],{},[454,8016,8017,8019],{},[364,8018,459],{"mathvariant":458},[364,8020,366],{},[368,8022,8023],{"encoding":370},"\\mathbb{R}^N",[344,8025,8027],{"className":8026,"ariaHidden":375},[374],[344,8028,8030,8033],{"className":8029},[379],[344,8031],{"className":8032,"style":502},[383],[344,8034,8036,8039],{"className":8035},[388],[344,8037,459],{"className":8038},[388,509],[344,8040,8042],{"className":8041},[513],[344,8043,8045],{"className":8044},[517],[344,8046,8048],{"className":8047},[521],[344,8049,8051],{"className":8050,"style":502},[525],[344,8052,8053,8056],{"style":528},[344,8054],{"className":8055,"style":533},[532],[344,8057,8059],{"className":8058},[537,538,539,540],[344,8060,366],{"className":8061,"style":390},[388,389,540]," 的每个紧子集中，只有有限多个临界点，因此临界点集具有 Lebesgue 测度零。",[15,8064,8065,8069,8070,8208,8209,8278,8279,1702,8404,8473,8474,8599,8600,800,8634,8703,8704,8773,8774,8807,8808],{},[8066,8067,8068],"em",{},"证明","：临界点是梯度映射 ",[344,8071,8073,8105],{"className":8072,"translate":348},[347],[344,8074,8076],{"className":8075},[352],[354,8077,8078],{"xmlns":356},[358,8079,8080,8102],{},[361,8081,8082,8084,8086,8088,8094,8096],{},[364,8083,1064],{"mathvariant":1063},[364,8085,448],{},[450,8087,452],{},[454,8089,8090,8092],{},[364,8091,459],{"mathvariant":458},[364,8093,366],{},[450,8095,464],{},[454,8097,8098,8100],{},[364,8099,459],{"mathvariant":458},[364,8101,366],{},[368,8103,8104],{"encoding":370},"\\nabla f: \\mathbb{R}^N \\to \\mathbb{R}^N",[344,8106,8108,8129,8173],{"className":8107,"ariaHidden":375},[374],[344,8109,8111,8114,8117,8120,8123,8126],{"className":8110},[379],[344,8112],{"className":8113,"style":479},[383],[344,8115,1064],{"className":8116},[388],[344,8118,448],{"className":8119,"style":483},[388,389],[344,8121],{"className":8122,"style":488},[487],[344,8124,452],{"className":8125},[492],[344,8127],{"className":8128,"style":488},[487],[344,8130,8132,8135,8164,8167,8170],{"className":8131},[379],[344,8133],{"className":8134,"style":502},[383],[344,8136,8138,8141],{"className":8137},[388],[344,8139,459],{"className":8140},[388,509],[344,8142,8144],{"className":8143},[513],[344,8145,8147],{"className":8146},[517],[344,8148,8150],{"className":8149},[521],[344,8151,8153],{"className":8152,"style":502},[525],[344,8154,8155,8158],{"style":528},[344,8156],{"className":8157,"style":533},[532],[344,8159,8161],{"className":8160},[537,538,539,540],[344,8162,366],{"className":8163,"style":390},[388,389,540],[344,8165],{"className":8166,"style":488},[487],[344,8168,464],{"className":8169},[492],[344,8171],{"className":8172,"style":488},[487],[344,8174,8176,8179],{"className":8175},[379],[344,8177],{"className":8178,"style":502},[383],[344,8180,8182,8185],{"className":8181},[388],[344,8183,459],{"className":8184},[388,509],[344,8186,8188],{"className":8187},[513],[344,8189,8191],{"className":8190},[517],[344,8192,8194],{"className":8193},[521],[344,8195,8197],{"className":8196,"style":502},[525],[344,8198,8199,8202],{"style":528},[344,8200],{"className":8201,"style":533},[532],[344,8203,8205],{"className":8204},[537,538,539,540],[344,8206,366],{"className":8207,"style":390},[388,389,540]," 的零点。该映射在 ",[344,8210,8212,8229],{"className":8211,"translate":348},[347],[344,8213,8215],{"className":8214},[352],[354,8216,8217],{"xmlns":356},[358,8218,8219,8227],{},[361,8220,8221],{},[822,8222,8223,8225],{},[364,8224,815],{},[364,8226,1721],{},[368,8228,4159],{"encoding":370},[344,8230,8232],{"className":8231,"ariaHidden":375},[374],[344,8233,8235,8238],{"className":8234},[379],[344,8236],{"className":8237,"style":3429},[383],[344,8239,8241,8244],{"className":8240},[388],[344,8242,815],{"className":8243},[388,389],[344,8245,8247],{"className":8246},[513],[344,8248,8250,8270],{"className":8249},[517,900],[344,8251,8253,8267],{"className":8252},[521],[344,8254,8256],{"className":8255,"style":1761},[525],[344,8257,8258,8261],{"style":910},[344,8259],{"className":8260,"style":533},[532],[344,8262,8264],{"className":8263},[537,538,539,540],[344,8265,1721],{"className":8266},[388,389,540],[344,8268,924],{"className":8269},[923],[344,8271,8273],{"className":8272},[521],[344,8274,8276],{"className":8275,"style":931},[525],[344,8277],{}," 处的导数就是 Hessian ",[344,8280,8282,8309],{"className":8281,"translate":348},[347],[344,8283,8285],{"className":8284},[352],[354,8286,8287],{"xmlns":356},[358,8288,8289,8307],{},[361,8290,8291,8297,8299,8305],{},[822,8292,8293,8295],{},[364,8294,2175],{},[364,8296,448],{},[450,8298,593],{"stretchy":592},[822,8300,8301,8303],{},[364,8302,815],{},[364,8304,1721],{},[450,8306,602],{"stretchy":592},[368,8308,4739],{"encoding":370},[344,8310,8312],{"className":8311,"ariaHidden":375},[374],[344,8313,8315,8318,8358,8361,8401],{"className":8314},[379],[344,8316],{"className":8317,"style":2281},[383],[344,8319,8321,8324],{"className":8320},[388],[344,8322,2175],{"className":8323,"style":2288},[388,389],[344,8325,8327],{"className":8326},[513],[344,8328,8330,8350],{"className":8329},[517,900],[344,8331,8333,8347],{"className":8332},[521],[344,8334,8336],{"className":8335,"style":2301},[525],[344,8337,8338,8341],{"style":2304},[344,8339],{"className":8340,"style":533},[532],[344,8342,8344],{"className":8343},[537,538,539,540],[344,8345,448],{"className":8346,"style":483},[388,389,540],[344,8348,924],{"className":8349},[923],[344,8351,8353],{"className":8352},[521],[344,8354,8356],{"className":8355,"style":2323},[525],[344,8357],{},[344,8359,593],{"className":8360},[668],[344,8362,8364,8367],{"className":8363},[388],[344,8365,815],{"className":8366},[388,389],[344,8368,8370],{"className":8369},[513],[344,8371,8373,8393],{"className":8372},[517,900],[344,8374,8376,8390],{"className":8375},[521],[344,8377,8379],{"className":8378,"style":1761},[525],[344,8380,8381,8384],{"style":910},[344,8382],{"className":8383,"style":533},[532],[344,8385,8387],{"className":8386},[537,538,539,540],[344,8388,1721],{"className":8389},[388,389,540],[344,8391,924],{"className":8392},[923],[344,8394,8396],{"className":8395},[521],[344,8397,8399],{"className":8398,"style":931},[525],[344,8400],{},[344,8402,602],{"className":8403},[701],[344,8405,8407,8424],{"className":8406,"translate":348},[347],[344,8408,8410],{"className":8409},[352],[354,8411,8412],{"xmlns":356},[358,8413,8414,8422],{},[361,8415,8416],{},[822,8417,8418,8420],{},[364,8419,815],{},[364,8421,1721],{},[368,8423,4159],{"encoding":370},[344,8425,8427],{"className":8426,"ariaHidden":375},[374],[344,8428,8430,8433],{"className":8429},[379],[344,8431],{"className":8432,"style":3429},[383],[344,8434,8436,8439],{"className":8435},[388],[344,8437,815],{"className":8438},[388,389],[344,8440,8442],{"className":8441},[513],[344,8443,8445,8465],{"className":8444},[517,900],[344,8446,8448,8462],{"className":8447},[521],[344,8449,8451],{"className":8450,"style":1761},[525],[344,8452,8453,8456],{"style":910},[344,8454],{"className":8455,"style":533},[532],[344,8457,8459],{"className":8458},[537,538,539,540],[344,8460,1721],{"className":8461},[388,389,540],[344,8463,924],{"className":8464},[923],[344,8466,8468],{"className":8467},[521],[344,8469,8471],{"className":8470,"style":931},[525],[344,8472],{}," 非退化，则 ",[344,8475,8477,8504],{"className":8476,"translate":348},[347],[344,8478,8480],{"className":8479},[352],[354,8481,8482],{"xmlns":356},[358,8483,8484,8502],{},[361,8485,8486,8492,8494,8500],{},[822,8487,8488,8490],{},[364,8489,2175],{},[364,8491,448],{},[450,8493,593],{"stretchy":592},[822,8495,8496,8498],{},[364,8497,815],{},[364,8499,1721],{},[450,8501,602],{"stretchy":592},[368,8503,4739],{"encoding":370},[344,8505,8507],{"className":8506,"ariaHidden":375},[374],[344,8508,8510,8513,8553,8556,8596],{"className":8509},[379],[344,8511],{"className":8512,"style":2281},[383],[344,8514,8516,8519],{"className":8515},[388],[344,8517,2175],{"className":8518,"style":2288},[388,389],[344,8520,8522],{"className":8521},[513],[344,8523,8525,8545],{"className":8524},[517,900],[344,8526,8528,8542],{"className":8527},[521],[344,8529,8531],{"className":8530,"style":2301},[525],[344,8532,8533,8536],{"style":2304},[344,8534],{"className":8535,"style":533},[532],[344,8537,8539],{"className":8538},[537,538,539,540],[344,8540,448],{"className":8541,"style":483},[388,389,540],[344,8543,924],{"className":8544},[923],[344,8546,8548],{"className":8547},[521],[344,8549,8551],{"className":8550,"style":2323},[525],[344,8552],{},[344,8554,593],{"className":8555},[668],[344,8557,8559,8562],{"className":8558},[388],[344,8560,815],{"className":8561},[388,389],[344,8563,8565],{"className":8564},[513],[344,8566,8568,8588],{"className":8567},[517,900],[344,8569,8571,8585],{"className":8570},[521],[344,8572,8574],{"className":8573,"style":1761},[525],[344,8575,8576,8579],{"style":910},[344,8577],{"className":8578,"style":533},[532],[344,8580,8582],{"className":8581},[537,538,539,540],[344,8583,1721],{"className":8584},[388,389,540],[344,8586,924],{"className":8587},[923],[344,8589,8591],{"className":8590},[521],[344,8592,8594],{"className":8593,"style":931},[525],[344,8595],{},[344,8597,602],{"className":8598},[701]," 可逆。由反函数定理，",[344,8601,8603,8619],{"className":8602,"translate":348},[347],[344,8604,8606],{"className":8605},[352],[354,8607,8608],{"xmlns":356},[358,8609,8610,8616],{},[361,8611,8612,8614],{},[364,8613,1064],{"mathvariant":1063},[364,8615,448],{},[368,8617,8618],{"encoding":370},"\\nabla f",[344,8620,8622],{"className":8621,"ariaHidden":375},[374],[344,8623,8625,8628,8631],{"className":8624},[379],[344,8626],{"className":8627,"style":479},[383],[344,8629,1064],{"className":8630},[388],[344,8632,448],{"className":8633,"style":483},[388,389],[344,8635,8637,8654],{"className":8636,"translate":348},[347],[344,8638,8640],{"className":8639},[352],[354,8641,8642],{"xmlns":356},[358,8643,8644,8652],{},[361,8645,8646],{},[822,8647,8648,8650],{},[364,8649,815],{},[364,8651,1721],{},[368,8653,4159],{"encoding":370},[344,8655,8657],{"className":8656,"ariaHidden":375},[374],[344,8658,8660,8663],{"className":8659},[379],[344,8661],{"className":8662,"style":3429},[383],[344,8664,8666,8669],{"className":8665},[388],[344,8667,815],{"className":8668},[388,389],[344,8670,8672],{"className":8671},[513],[344,8673,8675,8695],{"className":8674},[517,900],[344,8676,8678,8692],{"className":8677},[521],[344,8679,8681],{"className":8680,"style":1761},[525],[344,8682,8683,8686],{"style":910},[344,8684],{"className":8685,"style":533},[532],[344,8687,8689],{"className":8688},[537,538,539,540],[344,8690,1721],{"className":8691},[388,389,540],[344,8693,924],{"className":8694},[923],[344,8696,8698],{"className":8697},[521],[344,8699,8701],{"className":8700,"style":931},[525],[344,8702],{}," 附近局部可逆，因此 ",[344,8705,8707,8724],{"className":8706,"translate":348},[347],[344,8708,8710],{"className":8709},[352],[354,8711,8712],{"xmlns":356},[358,8713,8714,8722],{},[361,8715,8716],{},[822,8717,8718,8720],{},[364,8719,815],{},[364,8721,1721],{},[368,8723,4159],{"encoding":370},[344,8725,8727],{"className":8726,"ariaHidden":375},[374],[344,8728,8730,8733],{"className":8729},[379],[344,8731],{"className":8732,"style":3429},[383],[344,8734,8736,8739],{"className":8735},[388],[344,8737,815],{"className":8738},[388,389],[344,8740,8742],{"className":8741},[513],[344,8743,8745,8765],{"className":8744},[517,900],[344,8746,8748,8762],{"className":8747},[521],[344,8749,8751],{"className":8750,"style":1761},[525],[344,8752,8753,8756],{"style":910},[344,8754],{"className":8755,"style":533},[532],[344,8757,8759],{"className":8758},[537,538,539,540],[344,8760,1721],{"className":8761},[388,389,540],[344,8763,924],{"className":8764},[923],[344,8766,8768],{"className":8767},[521],[344,8769,8771],{"className":8770,"style":931},[525],[344,8772],{}," 是 ",[344,8775,8777,8792],{"className":8776,"translate":348},[347],[344,8778,8780],{"className":8779},[352],[354,8781,8782],{"xmlns":356},[358,8783,8784,8790],{},[361,8785,8786,8788],{},[364,8787,1064],{"mathvariant":1063},[364,8789,448],{},[368,8791,8618],{"encoding":370},[344,8793,8795],{"className":8794,"ariaHidden":375},[374],[344,8796,8798,8801,8804],{"className":8797},[379],[344,8799],{"className":8800,"style":479},[383],[344,8802,1064],{"className":8803},[388],[344,8805,448],{"className":8806,"style":483},[388,389]," 在某个邻域内的唯一零点。于是每个临界点都是孤立的。由于临界点集是闭集，紧区域内若有无限多个临界点，就会有一个同样为临界的聚点，这与孤立性矛盾。因此任何紧区域中只有有限多个临界点。有限集具有 Lebesgue 测度零。",[344,8809,8811,8826],{"className":8810,"translate":348},[347],[344,8812,8814],{"className":8813},[352],[354,8815,8816],{"xmlns":356},[358,8817,8818,8823],{},[361,8819,8820],{},[364,8821,8822],{"mathvariant":1063},"■",[368,8824,8825],{"encoding":370},"\\blacksquare",[344,8827,8829],{"className":8828,"ariaHidden":375},[374],[344,8830,8832,8836],{"className":8831},[379],[344,8833],{"className":8834,"style":8835},[383],"height:0.675em;",[344,8837,8822],{"className":8838},[388,8839],"amsrm",[15,8841,8842,8843,8901,8902,8905],{},"这说明：如果从 ",[344,8844,8846,8863],{"className":8845,"translate":348},[347],[344,8847,8849],{"className":8848},[352],[354,8850,8851],{"xmlns":356},[358,8852,8853,8861],{},[361,8854,8855],{},[454,8856,8857,8859],{},[364,8858,459],{"mathvariant":458},[364,8860,366],{},[368,8862,8023],{"encoding":370},[344,8864,8866],{"className":8865,"ariaHidden":375},[374],[344,8867,8869,8872],{"className":8868},[379],[344,8870],{"className":8871,"style":502},[383],[344,8873,8875,8878],{"className":8874},[388],[344,8876,459],{"className":8877},[388,509],[344,8879,8881],{"className":8880},[513],[344,8882,8884],{"className":8883},[517],[344,8885,8887],{"className":8886},[521],[344,8888,8890],{"className":8889,"style":502},[525],[344,8891,8892,8895],{"style":528},[344,8893],{"className":8894,"style":533},[532],[344,8896,8898],{"className":8897},[537,538,539,540],[344,8899,366],{"className":8900,"style":390},[388,389,540]," 中的一个盒子里均匀采样一点，恰好落在临界点上的概率为零。但真正有趣的问题是",[324,8903,8904],{},"条件性的","：",[764,8907,8908],{},[15,8909,8910],{},"既然我们已经身处某个高维随机函数的临界点，那么这个临界点是非退化局部极小值的概率是多少？",[15,8912,8913],{},"这个问题的答案，由 Hessian 特征值的符号来回答。",[11,8915,8916],{"id":8916},"马鞍面",[15,8918,8919],{},"在进入随机矩阵论证之前，先看一个最简单的鞍点例子，建立手感。",[15,8921,8922,8923,800,9008,9060],{},"函数 ",[344,8924,8926,8957],{"className":8925,"translate":348},[347],[344,8927,8929],{"className":8928},[352],[354,8930,8931],{"xmlns":356},[358,8932,8933,8954],{},[361,8934,8935,8937,8939,8941,8943,8946,8948,8950,8952],{},[364,8936,448],{},[450,8938,593],{"stretchy":592},[364,8940,815],{},[450,8942,831],{"separator":375},[364,8944,8945],{},"y",[450,8947,602],{"stretchy":592},[450,8949,818],{},[364,8951,815],{},[364,8953,8945],{},[368,8955,8956],{"encoding":370},"f(x,y) = xy",[344,8958,8960,8996],{"className":8959,"ariaHidden":375},[374],[344,8961,8963,8966,8969,8972,8975,8978,8981,8984,8987,8990,8993],{"className":8962},[379],[344,8964],{"className":8965,"style":884},[383],[344,8967,448],{"className":8968,"style":483},[388,389],[344,8970,593],{"className":8971},[668],[344,8973,815],{"className":8974},[388,389],[344,8976,831],{"className":8977},[937],[344,8979],{"className":8980,"style":941},[487],[344,8982,8945],{"className":8983,"style":4886},[388,389],[344,8985,602],{"className":8986},[701],[344,8988],{"className":8989,"style":488},[487],[344,8991,818],{"className":8992},[492],[344,8994],{"className":8995,"style":488},[487],[344,8997,8999,9002,9005],{"className":8998},[379],[344,9000],{"className":9001,"style":4879},[383],[344,9003,815],{"className":9004},[388,389],[344,9006,8945],{"className":9007,"style":4886},[388,389],[344,9009,9011,9033],{"className":9010,"translate":348},[347],[344,9012,9014],{"className":9013},[352],[354,9015,9016],{"xmlns":356},[358,9017,9018,9030],{},[361,9019,9020,9022,9024,9026,9028],{},[450,9021,593],{"stretchy":592},[587,9023,1096],{},[450,9025,831],{"separator":375},[587,9027,1096],{},[450,9029,602],{"stretchy":592},[368,9031,9032],{"encoding":370},"(0,0)",[344,9034,9036],{"className":9035,"ariaHidden":375},[374],[344,9037,9039,9042,9045,9048,9051,9054,9057],{"className":9038},[379],[344,9040],{"className":9041,"style":884},[383],[344,9043,593],{"className":9044},[668],[344,9046,1096],{"className":9047},[388],[344,9049,831],{"className":9050},[937],[344,9052],{"className":9053,"style":941},[487],[344,9055,1096],{"className":9056},[388],[344,9058,602],{"className":9059},[701]," 处有一个临界点，因为",[344,9062,9064],{"className":9063,"translate":348},[1047],[344,9065,9067,9110],{"className":9066,"translate":348},[347],[344,9068,9070],{"className":9069},[352],[354,9071,9072],{"xmlns":356,"display":1056},[358,9073,9074,9107],{},[361,9075,9076,9078,9080,9082,9084,9086,9088,9090,9092,9105],{},[364,9077,1064],{"mathvariant":1063},[364,9079,448],{},[450,9081,593],{"stretchy":592},[364,9083,815],{},[450,9085,831],{"separator":375},[364,9087,8945],{},[450,9089,602],{"stretchy":592},[450,9091,818],{},[361,9093,9094,9096,9103],{},[450,9095,593],{"fence":375},[1102,9097,9099,9101],{"linethickness":9098},"0px",[364,9100,8945],{},[364,9102,815],{},[450,9104,602],{"fence":375},[450,9106,831],{"separator":375},[368,9108,9109],{"encoding":370},"\\nabla f(x,y) = \\binom{y}{x},",[344,9111,9113,9152],{"className":9112,"ariaHidden":375},[374],[344,9114,9116,9119,9122,9125,9128,9131,9134,9137,9140,9143,9146,9149],{"className":9115},[379],[344,9117],{"className":9118,"style":884},[383],[344,9120,1064],{"className":9121},[388],[344,9123,448],{"className":9124,"style":483},[388,389],[344,9126,593],{"className":9127},[668],[344,9129,815],{"className":9130},[388,389],[344,9132,831],{"className":9133},[937],[344,9135],{"className":9136,"style":941},[487],[344,9138,8945],{"className":9139,"style":4886},[388,389],[344,9141,602],{"className":9142},[701],[344,9144],{"className":9145,"style":488},[487],[344,9147,818],{"className":9148},[492],[344,9150],{"className":9151,"style":488},[487],[344,9153,9155,9159,9220],{"className":9154},[379],[344,9156],{"className":9157,"style":9158},[383],"height:2.4em;vertical-align:-0.95em;",[344,9160,9162,9168,9214],{"className":9161},[388],[344,9163,9165],{"className":9164,"style":2419},[668,2418],[344,9166,593],{"className":9167},[1246,539],[344,9169,9171],{"className":9170},[1102],[344,9172,9174,9206],{"className":9173},[517,900],[344,9175,9177,9203],{"className":9176},[521],[344,9178,9181,9192],{"className":9179,"style":9180},[525],"height:1.1076em;",[344,9182,9183,9186],{"style":1348},[344,9184],{"className":9185,"style":1352},[532],[344,9187,9189],{"className":9188},[388],[344,9190,815],{"className":9191},[388,389],[344,9193,9194,9197],{"style":1413},[344,9195],{"className":9196,"style":1352},[532],[344,9198,9200],{"className":9199},[388],[344,9201,8945],{"className":9202,"style":4886},[388,389],[344,9204,924],{"className":9205},[923],[344,9207,9209],{"className":9208},[521],[344,9210,9212],{"className":9211,"style":3927},[525],[344,9213],{},[344,9215,9217],{"className":9216,"style":2419},[701,2418],[344,9218,602],{"className":9219},[1246,539],[344,9221,831],{"className":9222},[937],[15,9224,9225,9226,9310],{},"所以 ",[344,9227,9229,9259],{"className":9228,"translate":348},[347],[344,9230,9232],{"className":9231},[352],[354,9233,9234],{"xmlns":356},[358,9235,9236,9256],{},[361,9237,9238,9240,9242,9244,9246,9248,9250,9252,9254],{},[364,9239,1064],{"mathvariant":1063},[364,9241,448],{},[450,9243,593],{"stretchy":592},[587,9245,1096],{},[450,9247,831],{"separator":375},[587,9249,1096],{},[450,9251,602],{"stretchy":592},[450,9253,818],{},[587,9255,1096],{},[368,9257,9258],{"encoding":370},"\\nabla f(0,0) = 0",[344,9260,9262,9301],{"className":9261,"ariaHidden":375},[374],[344,9263,9265,9268,9271,9274,9277,9280,9283,9286,9289,9292,9295,9298],{"className":9264},[379],[344,9266],{"className":9267,"style":884},[383],[344,9269,1064],{"className":9270},[388],[344,9272,448],{"className":9273,"style":483},[388,389],[344,9275,593],{"className":9276},[668],[344,9278,1096],{"className":9279},[388],[344,9281,831],{"className":9282},[937],[344,9284],{"className":9285,"style":941},[487],[344,9287,1096],{"className":9288},[388],[344,9290,602],{"className":9291},[701],[344,9293],{"className":9294,"style":488},[487],[344,9296,818],{"className":9297},[492],[344,9299],{"className":9300,"style":488},[487],[344,9302,9304,9307],{"className":9303},[379],[344,9305],{"className":9306,"style":7133},[383],[344,9308,1096],{"className":9309},[388],"。它的 Hessian 是",[344,9312,9314],{"className":9313,"translate":348},[1047],[344,9315,9317,9386],{"className":9316,"translate":348},[347],[344,9318,9320],{"className":9319},[352],[354,9321,9322],{"xmlns":356,"display":1056},[358,9323,9324,9383],{},[361,9325,9326,9332,9334,9336,9338,9340,9342,9344,9381],{},[822,9327,9328,9330],{},[364,9329,2175],{},[364,9331,448],{},[450,9333,593],{"stretchy":592},[587,9335,1096],{},[450,9337,831],{"separator":375},[587,9339,1096],{},[450,9341,602],{"stretchy":592},[450,9343,818],{},[361,9345,9346,9348,9379],{},[450,9347,593],{"fence":375},[1082,9349,9351,9365],{"rowspacing":1084,"columnalign":9350,"columnspacing":1086},"center 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",[344,9599,9601,9614],{"className":9600,"translate":348},[347],[344,9602,9604],{"className":9603},[352],[354,9605,9606],{"xmlns":356},[358,9607,9608,9612],{},[361,9609,9610],{},[587,9611,828],{},[368,9613,828],{"encoding":370},[344,9615,9617],{"className":9616,"ariaHidden":375},[374],[344,9618,9620,9623],{"className":9619},[379],[344,9621],{"className":9622,"style":7133},[383],[344,9624,828],{"className":9625},[388]," 和 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",[344,9667,9669,9691],{"className":9668,"translate":348},[347],[344,9670,9672],{"className":9671},[352],[354,9673,9674],{"xmlns":356},[358,9675,9676,9688],{},[361,9677,9678,9680,9682,9684,9686],{},[450,9679,593],{"stretchy":592},[364,9681,815],{},[450,9683,831],{"separator":375},[364,9685,8945],{},[450,9687,602],{"stretchy":592},[368,9689,9690],{"encoding":370},"(x,y)",[344,9692,9694],{"className":9693,"ariaHidden":375},[374],[344,9695,9697,9700,9703,9706,9709,9712,9715],{"className":9696},[379],[344,9698],{"className":9699,"style":884},[383],[344,9701,593],{"className":9702},[668],[344,9704,815],{"className":9705},[388,389],[344,9707,831],{"className":9708},[937],[344,9710],{"className":9711,"style":941},[487],[344,9713,8945],{"className":9714,"style":4886},[388,389],[344,9716,602],{"className":9717},[701]," 坐标中，二阶导数 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与 ",[344,9800,9802,9824],{"className":9801,"translate":348},[347],[344,9803,9805],{"className":9804},[352],[354,9806,9807],{"xmlns":356},[358,9808,9809,9821],{},[361,9810,9811],{},[822,9812,9813,9815],{},[364,9814,448],{},[361,9816,9817,9819],{},[364,9818,8945],{},[364,9820,8945],{},[368,9822,9823],{"encoding":370},"f_{yy}",[344,9825,9827],{"className":9826,"ariaHidden":375},[374],[344,9828,9830,9834],{"className":9829},[379],[344,9831],{"className":9832,"style":9833},[383],"height:0.9805em;vertical-align:-0.2861em;",[344,9835,9837,9840],{"className":9836},[388],[344,9838,448],{"className":9839,"style":483},[388,389],[344,9841,9843],{"className":9842},[513],[344,9844,9846,9872],{"className":9845},[517,900],[344,9847,9849,9869],{"className":9848},[521],[344,9850,9852],{"className":9851,"style":1761},[525],[344,9853,9854,9857],{"style":9773},[344,9855],{"className":9856,"style":533},[532],[344,9858,9860],{"className":9859},[537,538,539,540],[344,9861,9863,9866],{"className":9862},[388,540],[344,9864,8945],{"className":9865,"style":4886},[388,389,540],[344,9867,8945],{"className":9868,"style":4886},[388,389,540],[344,9870,924],{"className":9871},[923],[344,9873,9875],{"className":9874},[521],[344,9876,9878],{"className":9877,"style":2323},[525],[344,9879],{}," 都为零，而混合导数 ",[344,9882,9884,9906],{"className":9883,"translate":348},[347],[344,9885,9887],{"className":9886},[352],[354,9888,9889],{"xmlns":356},[358,9890,9891,9903],{},[361,9892,9893],{},[822,9894,9895,9897],{},[364,9896,448],{},[361,9898,9899,9901],{},[364,9900,815],{},[364,9902,8945],{},[368,9904,9905],{"encoding":370},"f_{xy}",[344,9907,9909],{"className":9908,"ariaHidden":375},[374],[344,9910,9912,9915],{"className":9911},[379],[344,9913],{"className":9914,"style":9833},[383],[344,9916,9918,9921],{"className":9917},[388],[344,9919,448],{"className":9920,"style":483},[388,389],[344,9922,9924],{"className":9923},[513],[344,9925,9927,9953],{"className":9926},[517,900],[344,9928,9930,9950],{"className":9929},[521],[344,9931,9933],{"className":9932,"style":1761},[525],[344,9934,9935,9938],{"style":9773},[344,9936],{"className":9937,"style":533},[532],[344,9939,9941],{"className":9940},[537,538,539,540],[344,9942,9944,9947],{"className":9943},[388,540],[344,9945,815],{"className":9946},[388,389,540],[344,9948,8945],{"className":9949,"style":4886},[388,389,540],[344,9951,924],{"className":9952},[923],[344,9954,9956],{"className":9955},[521],[344,9957,9959],{"className":9958,"style":2323},[525],[344,9960],{}," 非零——主曲率方向并不是坐标轴。定义旋转后的坐标",[344,9963,9965],{"className":9964,"translate":348},[1047],[344,9966,9968,10025],{"className":9967,"translate":348},[347],[344,9969,9971],{"className":9970},[352],[354,9972,9973],{"xmlns":356,"display":1056},[358,9974,9975,10022],{},[361,9976,9977,9980,9982,9997,9999,10002,10004,10006,10020],{},[364,9978,9979],{},"u",[450,9981,818],{},[1102,9983,9984,9992],{},[361,9985,9986,9988,9990],{},[364,9987,815],{},[450,9989,3508],{},[364,9991,8945],{},[9993,9994,9995],"msqrt",{},[587,9996,589],{},[450,9998,831],{"separator":375},[487,10000],{"width":10001},"2em",[364,10003,4852],{},[450,10005,818],{},[1102,10007,10008,10016],{},[361,10009,10010,10012,10014],{},[364,10011,815],{},[450,10013,3372],{},[364,10015,8945],{},[9993,10017,10018],{},[587,10019,589],{},[450,10021,831],{"separator":375},[368,10023,10024],{"encoding":370},"u = \\frac{x+y}{\\sqrt2}, \\qquad v = 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= \\frac{u+v}{\\sqrt2}\\cdot\\frac{u-v}{\\sqrt2} = 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41,818],{"className":11042},[492],[344,11044],{"className":11045,"style":488},[487],[344,11047,11049,11053,11179],{"className":11048},[379],[344,11050],{"className":11051,"style":11052},[383],"height:2.1771em;vertical-align:-0.686em;",[344,11054,11056,11059,11176],{"className":11055},[388],[344,11057],{"className":11058},[668,1332],[344,11060,11062],{"className":11061},[1102],[344,11063,11065,11168],{"className":11064},[517,900],[344,11066,11068,11165],{"className":11067},[521],[344,11069,11071,11082,11090],{"className":11070,"style":2441},[525],[344,11072,11073,11076],{"style":1348},[344,11074],{"className":11075,"style":1352},[532],[344,11077,11079],{"className":11078},[388],[344,11080,589],{"className":11081},[388],[344,11083,11084,11087],{"style":1402},[344,11085],{"className":11086,"style":1352},[532],[344,11088],{"className":11089,"style":1410},[1409],[344,11091,11092,11095],{"style":1413},[344,11093],{"className":11094,"style":1352},[532],[344,11096,11098,11127,11130,11133,11136],{"className":11097},[388],[344,11099,11101,11104],{"className":11100},[388],[344,11102,9979],{"className":11103},[388,389],[344,11105,11107],{"className":11106},[513],[344,11108,11110],{"className":11109},[517],[344,11111,11113],{"className":11112},[521],[344,11114,11116],{"className":11115,"style":653},[525],[344,11117,11118,11121],{"style":528},[344,11119],{"className":11120,"style":533},[532],[344,11122,11124],{"className":11123},[537,538,539,540],[344,11125,589],{"className":11126},[388,540],[344,11128],{"className":11129,"style":2138},[487],[344,11131,3372],{"className":11132},[2142],[344,11134],{"className":11135,"style":2138},[487],[344,11137,11139,11142],{"className":11138},[388],[344,11140,4852],{"className":11141,"style":4886},[388,389],[344,11143,11145],{"className":11144},[513],[344,11146,11148],{"className":11147},[517],[344,11149,11151],{"className":11150},[521],[344,11152,11154],{"className":11153,"style":653},[525],[344,11155,11156,11159],{"style":528},[344,11157],{"className":11158,"style":533},[532],[344,11160,11162],{"className":11161},[537,538,539,540],[344,11163,589],{"className":11164},[388,540],[344,11166,924],{"className":11167},[923],[344,11169,11171],{"className":11170},[521],[344,11172,11174],{"className":11173,"style":3927},[525],[344,11175],{},[344,11177],{"className":11178},[701,1332],[344,11180,1192],{"className":11181},[388],[15,11183,11184],{},"在旋转后的坐标中，Hessian 是对角的：",[344,11186,11188],{"className":11187,"translate":348},[1047],[344,11189,11191,11249],{"className":11190,"translate":348},[347],[344,11192,11194],{"className":11193},[352],[354,11195,11196],{"xmlns":356,"display":1056},[358,11197,11198,11246],{},[361,11199,11200,11202,11204,11244],{},[364,11201,2175],{},[450,11203,818],{},[361,11205,11206,11208,11242],{},[450,11207,593],{"fence":375},[1082,11209,11210,11224],{"rowspacing":1084,"columnalign":9350,"columnspacing":1086},[1088,11211,11212,11218],{},[1091,11213,11214],{},[1094,11215,11216],{"scriptlevel":1096,"displaystyle":592},[587,11217,828],{},[1091,11219,11220],{},[1094,11221,11222],{"scriptlevel":1096,"displaystyle":592},[587,11223,1096],{},[1088,11225,11226,11232],{},[1091,11227,11228],{},[1094,11229,11230],{"scriptlevel":1096,"displaystyle":592},[587,11231,1096],{},[1091,11233,11234],{},[1094,11235,11236],{"scriptlevel":1096,"displaystyle":592},[361,11237,11238,11240],{},[450,11239,3372],{},[587,11241,828],{},[450,11243,602],{"fence":375},[364,11245,1192],{"mathvariant":1063},[368,11247,11248],{"encoding":370},"H = \\begin{pmatrix} 1 & 0 \\\\ 0 & -1 \\end{pmatrix}.",[344,11250,11252,11270],{"className":11251,"ariaHidden":375},[374],[344,11253,11255,11258,11261,11264,11267],{"className":11254},[379],[344,11256],{"className":11257,"style":384},[383],[344,11259,2175],{"className":11260,"style":2288},[388,389],[344,11262],{"className":11263,"style":488},[487],[344,11265,818],{"className":11266},[492],[344,11268],{"className":11269,"style":488},[487],[344,11271,11273,11276,11396,11399],{"className":11272},[379],[344,11274],{"className":11275,"style":9158},[383],[344,11277,11279,11285,11390],{"className":11278},[945],[344,11280,11282],{"className":11281,"style":2419},[668,2418],[344,11283,593],{"className":11284},[1246,539],[344,11286,11288],{"className":11287},[388],[344,11289,11291,11336,11339,11342],{"className":11290},[1082],[344,11292,11294],{"className":11293},[1305],[344,11295,11297,11328],{"className":11296},[517,900],[344,11298,11300,11325],{"className":11299},[521],[344,11301,11303,11314],{"className":11302,"style":9495},[525],[344,11304,11305,11308],{"style":9498},[344,11306],{"className":11307,"style":1352},[532],[344,11309,11311],{"className":11310},[388],[344,11312,828],{"className":11313},[388],[344,11315,11316,11319],{"style":9510},[344,11317],{"className":11318,"style":1352},[532],[344,11320,11322],{"className":11321},[388],[344,11323,1096],{"className":11324},[388],[344,11326,924],{"className":11327},[923],[344,11329,11331],{"className":11330},[521],[344,11332,11334],{"className":11333,"style":9529},[525],[344,11335],{},[344,11337],{"className":11338,"style":1301},[1300],[344,11340],{"className":11341,"style":1301},[1300],[344,11343,11345],{"className":11344},[1305],[344,11346,11348,11382],{"className":11347},[517,900],[344,11349,11351,11379],{"className":11350},[521],[344,11352,11354,11365],{"className":11353,"style":9495},[525],[344,11355,11356,11359],{"style":9498},[344,11357],{"className":11358,"style":1352},[532],[344,11360,11362],{"className":11361},[388],[344,11363,1096],{"className":11364},[388],[344,11366,11367,11370],{"style":9510},[344,11368],{"className":11369,"style":1352},[532],[344,11371,11373,11376],{"className":11372},[388],[344,11374,3372],{"className":11375},[388],[344,11377,828],{"className":11378},[388],[344,11380,924],{"className":11381},[923],[344,11383,11385],{"className":11384},[521],[344,11386,11388],{"className":11387,"style":9529},[525],[344,11389],{},[344,11391,11393],{"className":11392,"style":2419},[701,2418],[344,11394,602],{"className":11395},[1246,539],[344,11397],{"className":11398,"style":941},[487],[344,11400,1192],{"className":11401},[388],[15,11403,11404,11405,11433,11434,11462],{},"此刻鞍点结构一目了然：函数在 ",[344,11406,11408,11421],{"className":11407,"translate":348},[347],[344,11409,11411],{"className":11410},[352],[354,11412,11413],{"xmlns":356},[358,11414,11415,11419],{},[361,11416,11417],{},[364,11418,9979],{},[368,11420,9979],{"encoding":370},[344,11422,11424],{"className":11423,"ariaHidden":375},[374],[344,11425,11427,11430],{"className":11426},[379],[344,11428],{"className":11429,"style":865},[383],[344,11431,9979],{"className":11432},[388,389]," 方向向上弯曲，在 ",[344,11435,11437,11450],{"className":11436,"translate":348},[347],[344,11438,11440],{"className":11439},[352],[354,11441,11442],{"xmlns":356},[358,11443,11444,11448],{},[361,11445,11446],{},[364,11447,4852],{},[368,11449,4852],{"encoding":370},[344,11451,11453],{"className":11452,"ariaHidden":375},[374],[344,11454,11456,11459],{"className":11455},[379],[344,11457],{"className":11458,"style":865},[383],[344,11460,4852],{"className":11461,"style":4886},[388,389]," 方向向下弯曲。",[15,11464,11465,11466,11469,11470,11473],{},"不变的事实并不是 Hessian 在某个选定坐标系中的具体矩阵元素，而是它的",[324,11467,11468],{},"符号分布","：一个正特征值、一个负特征值。这是",[324,11471,11472],{},"西尔维斯特惯性定律（Sylvester's law of inertia）"," 的一个特例——该定律说，实二次型的正、负、零方向的数目，在可逆线性坐标变换下保持不变。",[11,11475,11476],{"id":11476},"随机矩阵的视角",[15,11478,11479],{},"现在回到主线问题：假设我们随机站在高维景观的某个临界点上，它是局部极小值的概率有多大？",[15,11481,11482],{},"第一个启发式是把典型临界点处的 Hessian 建模为一个随机实对称矩阵。这个模型对每个随机函数并非字面上都正确——Hessian 可能与函数值、梯度或问题中其它结构相关——但它抓住了核心机制：如果 Hessian 没有强的正偏向，它的特征值谱就应当同时含有正负号。",[11484,11485,11486],"h3",{"id":11486},"简化启发",[15,11488,11489,11490,11524],{},"仅仅出于直觉，假设每个 Hessian 特征值取正的概率为 ",[344,11491,11493,11512],{"className":11492,"translate":348},[347],[344,11494,11496],{"className":11495},[352],[354,11497,11498],{"xmlns":356},[358,11499,11500,11509],{},[361,11501,11502,11504,11507],{},[587,11503,828],{},[364,11505,11506],{"mathvariant":1063},"\u002F",[587,11508,589],{},[368,11510,11511],{"encoding":370},"1\u002F2",[344,11513,11515],{"className":11514,"ariaHidden":375},[374],[344,11516,11518,11521],{"className":11517},[379],[344,11519],{"className":11520,"style":884},[383],[344,11522,11511],{"className":11523},[388],"，且这些符号相互独立。那么",[344,11526,11528],{"className":11527,"translate":348},[1047],[344,11529,11531,11601],{"className":11530,"translate":348},[347],[344,11532,11534],{"className":11533},[352],[354,11535,11536],{"xmlns":356,"display":1056},[358,11537,11538,11598],{},[361,11539,11540,11543,11545,11548,11550,11552,11554,11556,11562,11564,11566,11568,11570,11572,11578,11580,11582,11584,11586,11596],{},[364,11541,11542],{"mathvariant":458},"P",[450,11544,593],{"stretchy":592},[3609,11546,11547],{},"非退化局部极小值",[450,11549,602],{"stretchy":592},[450,11551,818],{},[364,11553,11542],{"mathvariant":458},[450,11555,593],{"stretchy":592},[822,11557,11558,11560],{},[364,11559,4997],{},[587,11561,828],{},[450,11563,7222],{},[587,11565,1096],{},[450,11567,831],{"separator":375},[450,11569,834],{},[450,11571,831],{"separator":375},[822,11573,11574,11576],{},[364,11575,4997],{},[364,11577,366],{},[450,11579,7222],{},[587,11581,1096],{},[450,11583,602],{"stretchy":592},[450,11585,818],{},[454,11587,11588,11590],{},[587,11589,589],{},[361,11591,11592,11594],{},[450,11593,3372],{},[364,11595,366],{},[364,11597,1192],{"mathvariant":1063},[368,11599,11600],{"encoding":370},"\\mathbb{P}(\\text{非退化局部极小值}) = \\mathbb{P}(\\lambda_1>0,\\dots,\\lambda_N>0) = 2^{-N}.",[344,11602,11604,11635,11696,11772,11793],{"className":11603,"ariaHidden":375},[374],[344,11605,11607,11610,11613,11616,11623,11626,11629,11632],{"className":11606},[379],[344,11608],{"className":11609,"style":884},[383],[344,11611,11542],{"className":11612},[388,509],[344,11614,593],{"className":11615},[668],[344,11617,11619],{"className":11618},[388,4123],[344,11620,11547],{"className":11621},[388,11622],"cjk_fallback",[344,11624,602],{"className":11625},[701],[344,11627],{"className":11628,"style":488},[487],[344,11630,818],{"className":11631},[492],[344,11633],{"className":11634,"style":488},[487],[344,11636,11638,11641,11644,11647,11687,11690,11693],{"className":11637},[379],[344,11639],{"className":11640,"style":884},[383],[344,11642,11542],{"className":11643},[388,509],[344,11645,593],{"className":11646},[668],[344,11648,11650,11653],{"className":11649},[388],[344,11651,4997],{"className":11652},[388,389],[344,11654,11656],{"className":11655},[513],[344,11657,11659,11679],{"className":11658},[517,900],[344,11660,11662,11676],{"className":11661},[521],[344,11663,11665],{"className":11664,"style":907},[525],[344,11666,11667,11670],{"style":910},[344,11668],{"className":11669,"style":533},[532],[344,11671,11673],{"className":11672},[537,538,539,540],[344,11674,828],{"className":11675},[388,540],[344,11677,924],{"className":11678},[923],[344,11680,11682],{"className":11681},[521],[344,11683,11685],{"className":11684,"style":931},[525],[344,11686],{},[344,11688],{"className":11689,"style":488},[487],[344,11691,7222],{"className":11692},[492],[344,11694],{"className":11695,"style":488},[487],[344,11697,11699,11702,11705,11708,11711,11714,11717,11720,11723,11763,11766,11769],{"className":11698},[379],[344,11700],{"className":11701,"style":479},[383],[344,11703,1096],{"className":11704},[388],[344,11706,831],{"className":11707},[937],[344,11709],{"className":11710,"style":941},[487],[344,11712,834],{"className":11713},[945],[344,11715],{"className":11716,"style":941},[487],[344,11718,831],{"className":11719},[937],[344,11721],{"className":11722,"style":941},[487],[344,11724,11726,11729],{"className":11725},[388],[344,11727,4997],{"className":11728},[388,389],[344,11730,11732],{"className":11731},[513],[344,11733,11735,11755],{"className":11734},[517,900],[344,11736,11738,11752],{"className":11737},[521],[344,11739,11741],{"className":11740,"style":973},[525],[344,11742,11743,11746],{"style":910},[344,11744],{"className":11745,"style":533},[532],[344,11747,11749],{"className":11748},[537,538,539,540],[344,11750,366],{"className":11751,"style":390},[388,389,540],[344,11753,924],{"className":11754},[923],[344,11756,11758],{"className":11757},[521],[344,11759,11761],{"className":11760,"style":931},[525],[344,11762],{},[344,11764],{"className":11765,"style":488},[487],[344,11767,7222],{"className":11768},[492],[344,11770],{"className":11771,"style":488},[487],[344,11773,11775,11778,11781,11784,11787,11790],{"className":11774},[379],[344,11776],{"className":11777,"style":884},[383],[344,11779,1096],{"className":11780},[388],[344,11782,602],{"className":11783},[701],[344,11785],{"className":11786,"style":488},[487],[344,11788,818],{"className":11789},[492],[344,11791],{"className":11792,"style":488},[487],[344,11794,11796,11799,11834],{"className":11795},[379],[344,11797],{"className":11798,"style":1661},[383],[344,11800,11802,11805],{"className":11801},[388],[344,11803,589],{"className":11804},[388],[344,11806,11808],{"className":11807},[513],[344,11809,11811],{"className":11810},[517],[344,11812,11814],{"className":11813},[521],[344,11815,11817],{"className":11816,"style":1661},[525],[344,11818,11819,11822],{"style":1682},[344,11820],{"className":11821,"style":533},[532],[344,11823,11825],{"className":11824},[537,538,539,540],[344,11826,11828,11831],{"className":11827},[388,540],[344,11829,3372],{"className":11830},[388,540],[344,11832,366],{"className":11833,"style":390},[388,389,540],[344,11835,1192],{"className":11836},[388],[15,11838,11839,11840,11868],{},"这已经在 ",[344,11841,11843,11856],{"className":11842,"translate":348},[347],[344,11844,11846],{"className":11845},[352],[354,11847,11848],{"xmlns":356},[358,11849,11850,11854],{},[361,11851,11852],{},[364,11853,366],{},[368,11855,366],{"encoding":370},[344,11857,11859],{"className":11858,"ariaHidden":375},[374],[344,11860,11862,11865],{"className":11861},[379],[344,11863],{"className":11864,"style":384},[383],[344,11866,366],{"className":11867,"style":390},[388,389]," 上指数衰减。独立性假设对对称随机矩阵的特征值并不成立——特征值彼此强烈排斥——但「正定很罕见」这一结论仍然正确，而且真正的随机矩阵估计还要更尖锐。",[11484,11870,11872],{"id":11871},"goe-hessian-与半圆律","GOE Hessian 与半圆律",[15,11874,11875,11878,11879,11929,11930,11982,11983,12029],{},[324,11876,11877],{},"高斯正交系综（GOE）"," 是中心化随机实对称矩阵的标准模型。非正式地说，",[344,11880,11882,11899],{"className":11881,"translate":348},[347],[344,11883,11885],{"className":11884},[352],[354,11886,11887],{"xmlns":356},[358,11888,11889,11897],{},[361,11890,11891,11893,11895],{},[364,11892,366],{},[450,11894,2116],{},[364,11896,366],{},[368,11898,2121],{"encoding":370},[344,11900,11902,11920],{"className":11901,"ariaHidden":375},[374],[344,11903,11905,11908,11911,11914,11917],{"className":11904},[379],[344,11906],{"className":11907,"style":2131},[383],[344,11909,366],{"className":11910,"style":390},[388,389],[344,11912],{"className":11913,"style":2138},[487],[344,11915,2116],{"className":11916},[2142],[344,11918],{"className":11919,"style":2138},[487],[344,11921,11923,11926],{"className":11922},[379],[344,11924],{"className":11925,"style":384},[383],[344,11927,366],{"className":11928,"style":390},[388,389]," 的 GOE 矩阵是其独立元素为高斯的对称矩阵，归一化方式使得特征值在 ",[344,11931,11933,11952],{"className":11932,"translate":348},[347],[344,11934,11936],{"className":11935},[352],[354,11937,11938],{"xmlns":356},[358,11939,11940,11949],{},[361,11941,11942,11944,11946],{},[364,11943,366],{},[450,11945,464],{},[364,11947,11948],{"mathvariant":1063},"∞",[368,11950,11951],{"encoding":370},"N\\to\\infty",[344,11953,11955,11973],{"className":11954,"ariaHidden":375},[374],[344,11956,11958,11961,11964,11967,11970],{"className":11957},[379],[344,11959],{"className":11960,"style":384},[383],[344,11962,366],{"className":11963,"style":390},[388,389],[344,11965],{"className":11966,"style":488},[487],[344,11968,464],{"className":11969},[492],[344,11971],{"className":11972,"style":488},[487],[344,11974,11976,11979],{"className":11975},[379],[344,11977],{"className":11978,"style":865},[383],[344,11980,11948],{"className":11981},[388]," 时保持在 ",[344,11984,11986,12007],{"className":11985,"translate":348},[347],[344,11987,11989],{"className":11988},[352],[354,11990,11991],{"xmlns":356},[358,11992,11993,12004],{},[361,11994,11995,11998,12000,12002],{},[364,11996,11997],{},"O",[450,11999,593],{"stretchy":592},[587,12001,828],{},[450,12003,602],{"stretchy":592},[368,12005,12006],{"encoding":370},"O(1)",[344,12008,12010],{"className":12009,"ariaHidden":375},[374],[344,12011,12013,12016,12020,12023,12026],{"className":12012},[379],[344,12014],{"className":12015,"style":884},[383],[344,12017,11997],{"className":12018,"style":12019},[388,389],"margin-right:0.0278em;",[344,12021,593],{"className":12022},[668],[344,12024,828],{"className":12025},[388],[344,12027,602],{"className":12028},[701]," 量级。",[15,12031,12032,12035,12036,12086,12087,12124,12125,702],{},[324,12033,12034],{},"维格纳半圆律（Wigner's semicircle law）"," 表明，中心化 GOE 矩阵的经验特征值分布在 ",[344,12037,12039,12056],{"className":12038,"translate":348},[347],[344,12040,12042],{"className":12041},[352],[354,12043,12044],{"xmlns":356},[358,12045,12046,12054],{},[361,12047,12048,12050,12052],{},[364,12049,366],{},[450,12051,464],{},[364,12053,11948],{"mathvariant":1063},[368,12055,11951],{"encoding":370},[344,12057,12059,12077],{"className":12058,"ariaHidden":375},[374],[344,12060,12062,12065,12068,12071,12074],{"className":12061},[379],[344,12063],{"className":12064,"style":384},[383],[344,12066,366],{"className":12067,"style":390},[388,389],[344,12069],{"className":12070,"style":488},[487],[344,12072,464],{"className":12073},[492],[344,12075],{"className":12076,"style":488},[487],[344,12078,12080,12083],{"className":12079},[379],[344,12081],{"className":12082,"style":865},[383],[344,12084,11948],{"className":12085},[388]," 时收敛到关于零点对称的确定性半圆分布。因此，典型的大 GOE 矩阵有大量正特征值和大量负特征值，它的指数通常接近 ",[344,12088,12090,12108],{"className":12089,"translate":348},[347],[344,12091,12093],{"className":12092},[352],[354,12094,12095],{"xmlns":356},[358,12096,12097,12105],{},[361,12098,12099,12101,12103],{},[364,12100,366],{},[364,12102,11506],{"mathvariant":1063},[587,12104,589],{},[368,12106,12107],{"encoding":370},"N\u002F2",[344,12109,12111],{"className":12110,"ariaHidden":375},[374],[344,12112,12114,12117,12120],{"className":12113},[379],[344,12115],{"className":12116,"style":884},[383],[344,12118,366],{"className":12119,"style":390},[388,389],[344,12121,12123],{"className":12122},[388],"\u002F2","，而不是接近 0 或 ",[344,12126,12128,12141],{"className":12127,"translate":348},[347],[344,12129,12131],{"className":12130},[352],[354,12132,12133],{"xmlns":356},[358,12134,12135,12139],{},[361,12136,12137],{},[364,12138,366],{},[368,12140,366],{"encoding":370},[344,12142,12144],{"className":12143,"ariaHidden":375},[374],[344,12145,12147,12150],{"className":12146},[379],[344,12148],{"className":12149,"style":384},[383],[344,12151,366],{"className":12152,"style":390},[388,389],[15,12154,12155],{},"然而，一个临界点要成为非退化局部极小值，Hessian 必须正定。在 GOE 模型中，这就是事件",[344,12157,12159],{"className":12158,"translate":348},[1047],[344,12160,12162,12193],{"className":12161,"translate":348},[347],[344,12163,12165],{"className":12164},[352],[354,12166,12167],{"xmlns":356,"display":1056},[358,12168,12169,12190],{},[361,12170,12171,12180,12182,12184,12186,12188],{},[822,12172,12173,12175,12178],{},[364,12174,4997],{},[364,12176,12177],{},"min",[450,12179,6843],{},[450,12181,593],{"stretchy":592},[364,12183,2175],{},[450,12185,602],{"stretchy":592},[450,12187,7222],{},[587,12189,7260],{},[368,12191,12192],{"encoding":370},"\\lambda_{\\min}(H) > 0.",[344,12194,12196,12275],{"className":12195,"ariaHidden":375},[374],[344,12197,12199,12202,12257,12260,12263,12266,12269,12272],{"className":12198},[379],[344,12200],{"className":12201,"style":884},[383],[344,12203,12205,12208],{"className":12204},[388],[344,12206,4997],{"className":12207},[388,389],[344,12209,12211],{"className":12210},[513],[344,12212,12214,12249],{"className":12213},[517,900],[344,12215,12217,12246],{"className":12216},[521],[344,12218,12221],{"className":12219,"style":12220},[525],"height:0.3175em;",[344,12222,12223,12226],{"style":910},[344,12224],{"className":12225,"style":533},[532],[344,12227,12229],{"className":12228},[537,538,539,540],[344,12230,12232],{"className":12231},[388,540],[344,12233,12235,12239,12242],{"className":12234},[5207,540],[344,12236,12238],{"className":12237},[540],"m",[344,12240,2234],{"className":12241},[540],[344,12243,12245],{"className":12244},[540],"n",[344,12247,924],{"className":12248},[923],[344,12250,12252],{"className":12251},[521],[344,12253,12255],{"className":12254,"style":931},[525],[344,12256],{},[344,12258,593],{"className":12259},[668],[344,12261,2175],{"className":12262,"style":2288},[388,389],[344,12264,602],{"className":12265},[701],[344,12267],{"className":12268,"style":488},[487],[344,12270,7222],{"className":12271},[492],[344,12273],{"className":12274,"style":488},[487],[344,12276,12278,12281],{"className":12277},[379],[344,12279],{"className":12280,"style":7133},[383],[344,12282,7260],{"className":12283},[388],[15,12285,12286],{},"这个事件迫使整个特征值谱——其自然的极限形状以零为中心——平移到正的一侧。这是一个大偏差事件。",[764,12288,12289,12519,12935],{},[15,12290,12291,5826,12294,8773,12365,12415,12416,12467,12468,12518],{},[324,12292,12293],{},"定理 4.1（GOE 正定呈指数级罕见）",[344,12295,12297,12315],{"className":12296,"translate":348},[347],[344,12298,12300],{"className":12299},[352],[354,12301,12302],{"xmlns":356},[358,12303,12304,12312],{},[361,12305,12306],{},[822,12307,12308,12310],{},[364,12309,2175],{},[364,12311,366],{},[368,12313,12314],{"encoding":370},"H_N",[344,12316,12318],{"className":12317,"ariaHidden":375},[374],[344,12319,12321,12325],{"className":12320},[379],[344,12322],{"className":12323,"style":12324},[383],"height:0.8333em;vertical-align:-0.15em;",[344,12326,12328,12331],{"className":12327},[388],[344,12329,2175],{"className":12330,"style":2288},[388,389],[344,12332,12334],{"className":12333},[513],[344,12335,12337,12357],{"className":12336},[517,900],[344,12338,12340,12354],{"className":12339},[521],[344,12341,12343],{"className":12342,"style":973},[525],[344,12344,12345,12348],{"style":2304},[344,12346],{"className":12347,"style":533},[532],[344,12349,12351],{"className":12350},[537,538,539,540],[344,12352,366],{"className":12353,"style":390},[388,389,540],[344,12355,924],{"className":12356},[923],[344,12358,12360],{"className":12359},[521],[344,12361,12363],{"className":12362,"style":931},[525],[344,12364],{},[344,12366,12368,12385],{"className":12367,"translate":348},[347],[344,12369,12371],{"className":12370},[352],[354,12372,12373],{"xmlns":356},[358,12374,12375,12383],{},[361,12376,12377,12379,12381],{},[364,12378,366],{},[450,12380,2116],{},[364,12382,366],{},[368,12384,2121],{"encoding":370},[344,12386,12388,12406],{"className":12387,"ariaHidden":375},[374],[344,12389,12391,12394,12397,12400,12403],{"className":12390},[379],[344,12392],{"className":12393,"style":2131},[383],[344,12395,366],{"className":12396,"style":390},[388,389],[344,12398],{"className":12399,"style":2138},[487],[344,12401,2116],{"className":12402},[2142],[344,12404],{"className":12405,"style":2138},[487],[344,12407,12409,12412],{"className":12408},[379],[344,12410],{"className":12411,"style":384},[383],[344,12413,366],{"className":12414,"style":390},[388,389]," 中心化 GOE 矩阵，采用通常的随机矩阵标度。则存在常数 ",[344,12417,12419,12437],{"className":12418,"translate":348},[347],[344,12420,12422],{"className":12421},[352],[354,12423,12424],{"xmlns":356},[358,12425,12426,12434],{},[361,12427,12428,12430,12432],{},[364,12429,1721],{},[450,12431,7222],{},[587,12433,1096],{},[368,12435,12436],{"encoding":370},"c>0",[344,12438,12440,12458],{"className":12439,"ariaHidden":375},[374],[344,12441,12443,12446,12449,12452,12455],{"className":12442},[379],[344,12444],{"className":12445,"style":2051},[383],[344,12447,1721],{"className":12448},[388,389],[344,12450],{"className":12451,"style":488},[487],[344,12453,7222],{"className":12454},[492],[344,12456],{"className":12457,"style":488},[487],[344,12459,12461,12464],{"className":12460},[379],[344,12462],{"className":12463,"style":7133},[383],[344,12465,1096],{"className":12466},[388],"，使得在 ",[344,12469,12471,12488],{"className":12470,"translate":348},[347],[344,12472,12474],{"className":12473},[352],[354,12475,12476],{"xmlns":356},[358,12477,12478,12486],{},[361,12479,12480,12482,12484],{},[364,12481,366],{},[450,12483,464],{},[364,12485,11948],{"mathvariant":1063},[368,12487,11951],{"encoding":370},[344,12489,12491,12509],{"className":12490,"ariaHidden":375},[374],[344,12492,12494,12497,12500,12503,12506],{"className":12493},[379],[344,12495],{"className":12496,"style":384},[383],[344,12498,366],{"className":12499,"style":390},[388,389],[344,12501],{"className":12502,"style":488},[487],[344,12504,464],{"className":12505},[492],[344,12507],{"className":12508,"style":488},[487],[344,12510,12512,12515],{"className":12511},[379],[344,12513],{"className":12514,"style":865},[383],[344,12516,11948],{"className":12517},[388]," 时",[344,12520,12522],{"className":12521,"translate":348},[1047],[344,12523,12525,12618],{"className":12524,"translate":348},[347],[344,12526,12528],{"className":12527},[352],[354,12529,12530],{"xmlns":356,"display":1056},[358,12531,12532,12615],{},[361,12533,12534,12536,12538,12544,12546,12548,12550,12552,12554,12562,12564,12570,12572,12574,12576,12578,12580,12583,12585,12587,12589,12591,12597,12599,12601,12603,12609,12611,12613],{},[364,12535,11542],{"mathvariant":458},[450,12537,593],{"stretchy":592},[822,12539,12540,12542],{},[364,12541,2175],{},[364,12543,366],{},[3609,12545,326],{},[450,12547,602],{"stretchy":592},[450,12549,818],{},[364,12551,11542],{"mathvariant":458},[450,12553,593],{"stretchy":592},[822,12555,12556,12558,12560],{},[364,12557,4997],{},[364,12559,12177],{},[450,12561,6843],{},[450,12563,593],{"stretchy":592},[822,12565,12566,12568],{},[364,12567,2175],{},[364,12569,366],{},[450,12571,602],{"stretchy":592},[450,12573,7222],{},[587,12575,1096],{},[450,12577,602],{"stretchy":592},[450,12579,818],{},[364,12581,12582],{},"exp",[450,12584,6843],{},[450,12586,6863],{"stretchy":592},[450,12588,3372],{},[364,12590,1721],{},[454,12592,12593,12595],{},[364,12594,366],{},[587,12596,589],{},[450,12598,3508],{},[364,12600,3586],{},[450,12602,593],{"stretchy":592},[454,12604,12605,12607],{},[364,12606,366],{},[587,12608,589],{},[450,12610,602],{"stretchy":592},[450,12612,6881],{"stretchy":592},[364,12614,1192],{"mathvariant":1063},[368,12616,12617],{"encoding":370},"\\mathbb{P}(H_N \\text{正定}) = \\mathbb{P}(\\lambda_{\\min}(H_N)>0) = \\exp\\{-cN^2 + o(N^2)\\}.",[344,12619,12621,12691,12810,12831,12887],{"className":12620,"ariaHidden":375},[374],[344,12622,12624,12627,12630,12633,12673,12679,12682,12685,12688],{"className":12623},[379],[344,12625],{"className":12626,"style":884},[383],[344,12628,11542],{"className":12629},[388,509],[344,12631,593],{"className":12632},[668],[344,12634,12636,12639],{"className":12635},[388],[344,12637,2175],{"className":12638,"style":2288},[388,389],[344,12640,12642],{"className":12641},[513],[344,12643,12645,12665],{"className":12644},[517,900],[344,12646,12648,12662],{"className":12647},[521],[344,12649,12651],{"className":12650,"style":973},[525],[344,12652,12653,12656],{"style":2304},[344,12654],{"className":12655,"style":533},[532],[344,12657,12659],{"className":12658},[537,538,539,540],[344,12660,366],{"className":12661,"style":390},[388,389,540],[344,12663,924],{"className":12664},[923],[344,12666,12668],{"className":12667},[521],[344,12669,12671],{"className":12670,"style":931},[525],[344,12672],{},[344,12674,12676],{"className":12675},[388,4123],[344,12677,326],{"className":12678},[388,11622],[344,12680,602],{"className":12681},[701],[344,12683],{"className":12684,"style":488},[487],[344,12686,818],{"className":12687},[492],[344,12689],{"className":12690,"style":488},[487],[344,12692,12694,12697,12700,12703,12755,12758,12798,12801,12804,12807],{"className":12693},[379],[344,12695],{"className":12696,"style":884},[383],[344,12698,11542],{"className":12699},[388,509],[344,12701,593],{"className":12702},[668],[344,12704,12706,12709],{"className":12705},[388],[344,12707,4997],{"className":12708},[388,389],[344,12710,12712],{"className":12711},[513],[344,12713,12715,12747],{"className":12714},[517,900],[344,12716,12718,12744],{"className":12717},[521],[344,12719,12721],{"className":12720,"style":12220},[525],[344,12722,12723,12726],{"style":910},[344,12724],{"className":12725,"style":533},[532],[344,12727,12729],{"className":12728},[537,538,539,540],[344,12730,12732],{"className":12731},[388,540],[344,12733,12735,12738,12741],{"className":12734},[5207,540],[344,12736,12238],{"className":12737},[540],[344,12739,2234],{"className":12740},[540],[344,12742,12245],{"className":12743},[540],[344,12745,924],{"className":12746},[923],[344,12748,12750],{"className":12749},[521],[344,12751,12753],{"className":12752,"style":931},[525],[344,12754],{},[344,12756,593],{"className":12757},[668],[344,12759,12761,12764],{"className":12760},[388],[344,12762,2175],{"className":12763,"style":2288},[388,389],[344,12765,12767],{"className":12766},[513],[344,12768,12770,12790],{"className":12769},[517,900],[344,12771,12773,12787],{"className":12772},[521],[344,12774,12776],{"className":12775,"style":973},[525],[344,12777,12778,12781],{"style":2304},[344,12779],{"className":12780,"style":533},[532],[344,12782,12784],{"className":12783},[537,538,539,540],[344,12785,366],{"className":12786,"style":390},[388,389,540],[344,12788,924],{"className":12789},[923],[344,12791,12793],{"className":12792},[521],[344,12794,12796],{"className":12795,"style":931},[525],[344,12797],{},[344,12799,602],{"className":12800},[701],[344,12802],{"className":12803,"style":488},[487],[344,12805,7222],{"className":12806},[492],[344,12808],{"className":12809,"style":488},[487],[344,12811,12813,12816,12819,12822,12825,12828],{"className":12812},[379],[344,12814],{"className":12815,"style":884},[383],[344,12817,1096],{"className":12818},[388],[344,12820,602],{"className":12821},[701],[344,12823],{"className":12824,"style":488},[487],[344,12826,818],{"className":12827},[492],[344,12829],{"className":12830,"style":488},[487],[344,12832,12834,12837,12840,12843,12846,12849,12878,12881,12884],{"className":12833},[379],[344,12835],{"className":12836,"style":2350},[383],[344,12838,12582],{"className":12839},[5207],[344,12841,6863],{"className":12842},[668],[344,12844,3372],{"className":12845},[388],[344,12847,1721],{"className":12848},[388,389],[344,12850,12852,12855],{"className":12851},[388],[344,12853,366],{"className":12854,"style":390},[388,389],[344,12856,12858],{"className":12857},[513],[344,12859,12861],{"className":12860},[517],[344,12862,12864],{"className":12863},[521],[344,12865,12867],{"className":12866,"style":2369},[525],[344,12868,12869,12872],{"style":1682},[344,12870],{"className":12871,"style":533},[532],[344,12873,12875],{"className":12874},[537,538,539,540],[344,12876,589],{"className":12877},[388,540],[344,12879],{"className":12880,"style":2138},[487],[344,12882,3508],{"className":12883},[2142],[344,12885],{"className":12886,"style":2138},[487],[344,12888,12890,12893,12896,12899,12928,12932],{"className":12889},[379],[344,12891],{"className":12892,"style":2350},[383],[344,12894,3586],{"className":12895},[388,389],[344,12897,593],{"className":12898},[668],[344,12900,12902,12905],{"className":12901},[388],[344,12903,366],{"className":12904,"style":390},[388,389],[344,12906,12908],{"className":12907},[513],[344,12909,12911],{"className":12910},[517],[344,12912,12914],{"className":12913},[521],[344,12915,12917],{"className":12916,"style":2369},[525],[344,12918,12919,12922],{"style":1682},[344,12920],{"className":12921,"style":533},[532],[344,12923,12925],{"className":12924},[537,538,539,540],[344,12926,589],{"className":12927},[388,540],[344,12929,12931],{"className":12930},[701],")}",[344,12933,1192],{"className":12934},[388],[15,12936,12937,12938,12969,12970,13067,13068,13170],{},"更精确地说，对高斯 ",[344,12939,12941,12956],{"className":12940,"translate":348},[347],[344,12942,12944],{"className":12943},[352],[354,12945,12946],{"xmlns":356},[358,12947,12948,12953],{},[361,12949,12950],{},[364,12951,12952],{},"β",[368,12954,12955],{"encoding":370},"\\beta",[344,12957,12959],{"className":12958,"ariaHidden":375},[374],[344,12960,12962,12965],{"className":12961},[379],[344,12963],{"className":12964,"style":479},[383],[344,12966,12952],{"className":12967,"style":12968},[388,389],"margin-right:0.0528em;"," 系综，指数具有 ",[344,12971,12973,13007],{"className":12972,"translate":348},[347],[344,12974,12976],{"className":12975},[352],[354,12977,12978],{"xmlns":356},[358,12979,12980,13004],{},[361,12981,12982,12984,12987,12990,12992,12994,12996,12998],{},[364,12983,12952],{},[3609,12985,12986],{}," ",[364,12988,12989],{},"θ",[450,12991,593],{"stretchy":592},[587,12993,1096],{},[450,12995,602],{"stretchy":592},[3609,12997,12986],{},[454,12999,13000,13002],{},[364,13001,366],{},[587,13003,589],{},[368,13005,13006],{"encoding":370},"\\beta\\,\\theta(0)\\,N^2",[344,13008,13010],{"className":13009,"ariaHidden":375},[374],[344,13011,13013,13017,13020,13023,13026,13029,13032,13035,13038],{"className":13012},[379],[344,13014],{"className":13015,"style":13016},[383],"height:1.0641em;vertical-align:-0.25em;",[344,13018,12952],{"className":13019,"style":12968},[388,389],[344,13021],{"className":13022,"style":941},[487],[344,13024,12989],{"className":13025,"style":12019},[388,389],[344,13027,593],{"className":13028},[668],[344,13030,1096],{"className":13031},[388],[344,13033,602],{"className":13034},[701],[344,13036],{"className":13037,"style":941},[487],[344,13039,13041,13044],{"className":13040},[388],[344,13042,366],{"className":13043,"style":390},[388,389],[344,13045,13047],{"className":13046},[513],[344,13048,13050],{"className":13049},[517],[344,13051,13053],{"className":13052},[521],[344,13054,13056],{"className":13055,"style":653},[525],[344,13057,13058,13061],{"style":528},[344,13059],{"className":13060,"style":533},[532],[344,13062,13064],{"className":13063},[537,538,539,540],[344,13065,589],{"className":13066},[388,540]," 的形式，其中 ",[344,13069,13071,13110],{"className":13070,"translate":348},[347],[344,13072,13074],{"className":13073},[352],[354,13075,13076],{"xmlns":356},[358,13077,13078,13107],{},[361,13079,13080,13082,13084,13086,13088,13090,13092,13095,13097,13100,13102,13104],{},[364,13081,12989],{},[450,13083,593],{"stretchy":592},[587,13085,1096],{},[450,13087,602],{"stretchy":592},[450,13089,818],{},[450,13091,593],{"stretchy":592},[364,13093,13094],{},"log",[450,13096,6843],{},[587,13098,13099],{},"3",[450,13101,602],{"stretchy":592},[364,13103,11506],{"mathvariant":1063},[587,13105,13106],{},"4",[368,13108,13109],{"encoding":370},"\\theta(0) = (\\log 3)\u002F4",[344,13111,13113,13140],{"className":13112,"ariaHidden":375},[374],[344,13114,13116,13119,13122,13125,13128,13131,13134,13137],{"className":13115},[379],[344,13117],{"className":13118,"style":884},[383],[344,13120,12989],{"className":13121,"style":12019},[388,389],[344,13123,593],{"className":13124},[668],[344,13126,1096],{"className":13127},[388],[344,13129,602],{"className":13130},[701],[344,13132],{"className":13133,"style":488},[487],[344,13135,818],{"className":13136},[492],[344,13138],{"className":13139,"style":488},[487],[344,13141,13143,13146,13149,13157,13160,13163,13166],{"className":13142},[379],[344,13144],{"className":13145,"style":884},[383],[344,13147,593],{"className":13148},[668],[344,13150,13152,13153],{"className":13151},[5207],"lo",[344,13154,13156],{"style":13155},"margin-right:0.0139em;","g",[344,13158],{"className":13159,"style":941},[487],[344,13161,13099],{"className":13162},[388],[344,13164,602],{"className":13165},[701],[344,13167,13169],{"className":13168},[388],"\u002F4","（Dean & Majumdar 的结果与约定）。",[15,13172,13173,13174,13233,13234,13237,13238,13307],{},"精确常数不如标度重要。要点在于：正定不仅是「不太可能」，对中心化 GOE 矩阵而言，它在 ",[344,13175,13177,13195],{"className":13176,"translate":348},[347],[344,13178,13180],{"className":13179},[352],[354,13181,13182],{"xmlns":356},[358,13183,13184,13192],{},[361,13185,13186],{},[454,13187,13188,13190],{},[364,13189,366],{},[587,13191,589],{},[368,13193,13194],{"encoding":370},"N^2",[344,13196,13198],{"className":13197,"ariaHidden":375},[374],[344,13199,13201,13204],{"className":13200},[379],[344,13202],{"className":13203,"style":653},[383],[344,13205,13207,13210],{"className":13206},[388],[344,13208,366],{"className":13209,"style":390},[388,389],[344,13211,13213],{"className":13212},[513],[344,13214,13216],{"className":13215},[517],[344,13217,13219],{"className":13218},[521],[344,13220,13222],{"className":13221,"style":653},[525],[344,13223,13224,13227],{"style":528},[344,13225],{"className":13226,"style":533},[532],[344,13228,13230],{"className":13229},[537,538,539,540],[344,13231,589],{"className":13232},[388,540]," 上呈",[324,13235,13236],{},"指数级","不可能——比前面粗糙估计里的 ",[344,13239,13241,13263],{"className":13240,"translate":348},[347],[344,13242,13244],{"className":13243},[352],[354,13245,13246],{"xmlns":356},[358,13247,13248,13260],{},[361,13249,13250],{},[454,13251,13252,13254],{},[587,13253,589],{},[361,13255,13256,13258],{},[450,13257,3372],{},[364,13259,366],{},[368,13261,13262],{"encoding":370},"2^{-N}",[344,13264,13266],{"className":13265,"ariaHidden":375},[374],[344,13267,13269,13272],{"className":13268},[379],[344,13270],{"className":13271,"style":502},[383],[344,13273,13275,13278],{"className":13274},[388],[344,13276,589],{"className":13277},[388],[344,13279,13281],{"className":13280},[513],[344,13282,13284],{"className":13283},[517],[344,13285,13287],{"className":13286},[521],[344,13288,13290],{"className":13289,"style":502},[525],[344,13291,13292,13295],{"style":528},[344,13293],{"className":13294,"style":533},[532],[344,13296,13298],{"className":13297},[537,538,539,540],[344,13299,13301,13304],{"className":13300},[388,540],[344,13302,3372],{"className":13303},[388,540],[344,13305,366],{"className":13306,"style":390},[388,389,540]," 还要猛烈。于是，在随机 Hessian 模型中，",[344,13309,13311],{"className":13310,"translate":348},[1047],[344,13312,13314,13357],{"className":13313,"translate":348},[347],[344,13315,13317],{"className":13316},[352],[354,13318,13319],{"xmlns":356,"display":1056},[358,13320,13321,13354],{},[361,13322,13323,13325,13327,13330,13332,13335,13337,13339,13341,13343,13345,13348,13350,13352],{},[364,13324,11542],{"mathvariant":458},[450,13326,593],{"stretchy":592},[3609,13328,13329],{},"临界点是鞍点",[450,13331,602],{"stretchy":592},[450,13333,13334],{},"⟶",[587,13336,828],{},[450,13338,831],{"separator":375},[487,13340],{"width":10001},[364,13342,11542],{"mathvariant":458},[450,13344,593],{"stretchy":592},[3609,13346,13347],{},"临界点是非退化局部极小值",[450,13349,602],{"stretchy":592},[450,13351,13334],{},[587,13353,7260],{},[368,13355,13356],{"encoding":370},"\\mathbb{P}(\\text{临界点是鞍点}) \\longrightarrow 1, \\qquad \\mathbb{P}(\\text{临界点是非退化局部极小值}) \\longrightarrow 0.",[344,13358,13360,13390,13432],{"className":13359,"ariaHidden":375},[374],[344,13361,13363,13366,13369,13372,13378,13381,13384,13387],{"className":13362},[379],[344,13364],{"className":13365,"style":884},[383],[344,13367,11542],{"className":13368},[388,509],[344,13370,593],{"className":13371},[668],[344,13373,13375],{"className":13374},[388,4123],[344,13376,13329],{"className":13377},[388,11622],[344,13379,602],{"className":13380},[701],[344,13382],{"className":13383,"style":488},[487],[344,13385,13334],{"className":13386},[492],[344,13388],{"className":13389,"style":488},[487],[344,13391,13393,13396,13399,13402,13405,13408,13411,13414,13420,13423,13426,13429],{"className":13392},[379],[344,13394],{"className":13395,"style":884},[383],[344,13397,828],{"className":13398},[388],[344,13400,831],{"className":13401},[937],[344,13403],{"className":13404,"style":10191},[487],[344,13406],{"className":13407,"style":941},[487],[344,13409,11542],{"className":13410},[388,509],[344,13412,593],{"className":13413},[668],[344,13415,13417],{"className":13416},[388,4123],[344,13418,13347],{"className":13419},[388,11622],[344,13421,602],{"className":13422},[701],[344,13424],{"className":13425,"style":488},[487],[344,13427,13334],{"className":13428},[492],[344,13430],{"className":13431,"style":488},[487],[344,13433,13435,13438],{"className":13434},[379],[344,13436],{"className":13437,"style":7133},[383],[344,13439,7260],{"className":13440},[388],[11484,13442,13443],{"id":13443},"更明确的几何诠释",[15,13445,4138,13446,13515],{},[344,13447,13449,13466],{"className":13448,"translate":348},[347],[344,13450,13452],{"className":13451},[352],[354,13453,13454],{"xmlns":356},[358,13455,13456,13464],{},[361,13457,13458],{},[822,13459,13460,13462],{},[364,13461,815],{},[364,13463,1721],{},[368,13465,4159],{"encoding":370},[344,13467,13469],{"className":13468,"ariaHidden":375},[374],[344,13470,13472,13475],{"className":13471},[379],[344,13473],{"className":13474,"style":3429},[383],[344,13476,13478,13481],{"className":13477},[388],[344,13479,815],{"className":13480},[388,389],[344,13482,13484],{"className":13483},[513],[344,13485,13487,13507],{"className":13486},[517,900],[344,13488,13490,13504],{"className":13489},[521],[344,13491,13493],{"className":13492,"style":1761},[525],[344,13494,13495,13498],{"style":910},[344,13496],{"className":13497,"style":533},[532],[344,13499,13501],{"className":13500},[537,538,539,540],[344,13502,1721],{"className":13503},[388,389,540],[344,13505,924],{"className":13506},[923],[344,13508,13510],{"className":13509},[521],[344,13511,13513],{"className":13512,"style":931},[525],[344,13514],{}," 处，二阶近似为",[344,13517,13519],{"className":13518,"translate":348},[1047],[344,13520,13522,13599],{"className":13521,"translate":348},[347],[344,13523,13525],{"className":13524},[352],[354,13526,13527],{"xmlns":356,"display":1056},[358,13528,13529,13596],{},[361,13530,13531,13533,13535,13541,13543,13545,13547,13549,13551,13553,13559,13561,13564,13570,13576,13582,13584,13590,13592,13594],{},[364,13532,448],{},[450,13534,593],{"stretchy":592},[822,13536,13537,13539],{},[364,13538,815],{},[364,13540,1721],{},[450,13542,3508],{},[364,13544,3365],{},[450,13546,602],{"stretchy":592},[450,13548,3372],{},[364,13550,448],{},[450,13552,593],{"stretchy":592},[822,13554,13555,13557],{},[364,13556,815],{},[364,13558,1721],{},[450,13560,602],{"stretchy":592},[450,13562,13563],{},"≈",[1102,13565,13566,13568],{},[587,13567,828],{},[587,13569,589],{},[454,13571,13572,13574],{},[364,13573,3365],{},[364,13575,3547],{},[822,13577,13578,13580],{},[364,13579,2175],{},[364,13581,448],{},[450,13583,593],{"stretchy":592},[822,13585,13586,13588],{},[364,13587,815],{},[364,13589,1721],{},[450,13591,602],{"stretchy":592},[364,13593,3365],{},[364,13595,1192],{"mathvariant":1063},[368,13597,13598],{"encoding":370},"f(x_c+h) - f(x_c) \\approx \\frac{1}{2}h^T H_f(x_c) h.",[344,13600,13602,13663,13684,13748],{"className":13601,"ariaHidden":375},[374],[344,13603,13605,13608,13611,13614,13654,13657,13660],{"className":13604},[379],[344,13606],{"className":13607,"style":884},[383],[344,13609,448],{"className":13610,"style":483},[388,389],[344,13612,593],{"className":13613},[668],[344,13615,13617,13620],{"className":13616},[388],[344,13618,815],{"className":13619},[388,389],[344,13621,13623],{"className":13622},[513],[344,13624,13626,13646],{"className":13625},[517,900],[344,13627,13629,13643],{"className":13628},[521],[344,13630,13632],{"className":13631,"style":1761},[525],[344,13633,13634,13637],{"style":910},[344,13635],{"className":13636,"style":533},[532],[344,13638,13640],{"className":13639},[537,538,539,540],[344,13641,1721],{"className":13642},[388,389,540],[344,13644,924],{"className":13645},[923],[344,13647,13649],{"className":13648},[521],[344,13650,13652],{"className":13651,"style":931},[525],[344,13653],{},[344,13655],{"className":13656,"style":2138},[487],[344,13658,3508],{"className":13659},[2142],[344,13661],{"className":13662,"style":2138},[487],[344,13664,13666,13669,13672,13675,13678,13681],{"className":13665},[379],[344,13667],{"className":13668,"style":884},[383],[344,13670,3365],{"className":13671},[388,389],[344,13673,602],{"className":13674},[701],[344,13676],{"className":13677,"style":2138},[487],[344,13679,3372],{"className":13680},[2142],[344,13682],{"className":13683,"style":2138},[487],[344,13685,13687,13690,13693,13696,13736,13739,13742,13745],{"className":13686},[379],[344,13688],{"className":13689,"style":884},[383],[344,13691,448],{"className":13692,"style":483},[388,389],[344,13694,593],{"className":13695},[668],[344,13697,13699,13702],{"className":13698},[388],[344,13700,815],{"className":13701},[388,389],[344,13703,13705],{"className":13704},[513],[344,13706,13708,13728],{"className":13707},[517,900],[344,13709,13711,13725],{"className":13710},[521],[344,13712,13714],{"className":13713,"style":1761},[525],[344,13715,13716,13719],{"style":910},[344,13717],{"className":13718,"style":533},[532],[344,13720,13722],{"className":13721},[537,538,539,540],[344,13723,1721],{"className":13724},[388,389,540],[344,13726,924],{"className":13727},[923],[344,13729,13731],{"className":13730},[521],[344,13732,13734],{"className":13733,"style":931},[525],[344,13735],{},[344,13737,602],{"className":13738},[701],[344,13740],{"className":13741,"style":488},[487],[344,13743,13563],{"className":13744},[492],[344,13746],{"className":13747,"style":488},[487],[344,13749,13751,13754,13816,13845,13885,13888,13928,13931,13934],{"className":13750},[379],[344,13752],{"className":13753,"style":3868},[383],[344,13755,13757,13760,13813],{"className":13756},[388],[344,13758],{"className":13759},[668,1332],[344,13761,13763],{"className":13762},[1102],[344,13764,13766,13805],{"className":13765},[517,900],[344,13767,13769,13802],{"className":13768},[521],[344,13770,13772,13783,13791],{"className":13771,"style":3887},[525],[344,13773,13774,13777],{"style":1348},[344,13775],{"className":13776,"style":1352},[532],[344,13778,13780],{"className":13779},[388],[344,13781,589],{"className":13782},[388],[344,13784,13785,13788],{"style":1402},[344,13786],{"className":13787,"style":1352},[532],[344,13789],{"className":13790,"style":1410},[1409],[344,13792,13793,13796],{"style":1413},[344,13794],{"className":13795,"style":1352},[532],[344,13797,13799],{"className":13798},[388],[344,13800,828],{"className":13801},[388],[344,13803,924],{"className":13804},[923],[344,13806,13808],{"className":13807},[521],[344,13809,13811],{"className":13810,"style":3927},[525],[344,13812],{},[344,13814],{"className":13815},[701,1332],[344,13817,13819,13822],{"className":13818},[388],[344,13820,3365],{"className":13821},[388,389],[344,13823,13825],{"className":13824},[513],[344,13826,13828],{"className":13827},[517],[344,13829,13831],{"className":13830},[521],[344,13832,13834],{"className":13833,"style":1661},[525],[344,13835,13836,13839],{"style":1682},[344,13837],{"className":13838,"style":533},[532],[344,13840,13842],{"className":13841},[537,538,539,540],[344,13843,3547],{"className":13844,"style":3849},[388,389,540],[344,13846,13848,13851],{"className":13847},[388],[344,13849,2175],{"className":13850,"style":2288},[388,389],[344,13852,13854],{"className":13853},[513],[344,13855,13857,13877],{"className":13856},[517,900],[344,13858,13860,13874],{"className":13859},[521],[344,13861,13863],{"className":13862,"style":2301},[525],[344,13864,13865,13868],{"style":2304},[344,13866],{"className":13867,"style":533},[532],[344,13869,13871],{"className":13870},[537,538,539,540],[344,13872,448],{"className":13873,"style":483},[388,389,540],[344,13875,924],{"className":13876},[923],[344,13878,13880],{"className":13879},[521],[344,13881,13883],{"className":13882,"style":2323},[525],[344,13884],{},[344,13886,593],{"className":13887},[668],[344,13889,13891,13894],{"className":13890},[388],[344,13892,815],{"className":13893},[388,389],[344,13895,13897],{"className":13896},[513],[344,13898,13900,13920],{"className":13899},[517,900],[344,13901,13903,13917],{"className":13902},[521],[344,13904,13906],{"className":13905,"style":1761},[525],[344,13907,13908,13911],{"style":910},[344,13909],{"className":13910,"style":533},[532],[344,13912,13914],{"className":13913},[537,538,539,540],[344,13915,1721],{"className":13916},[388,389,540],[344,13918,924],{"className":13919},[923],[344,13921,13923],{"className":13922},[521],[344,13924,13926],{"className":13925,"style":931},[525],[344,13927],{},[344,13929,602],{"className":13930},[701],[344,13932,3365],{"className":13933},[388,389],[344,13935,1192],{"className":13936},[388],[15,13938,6028,13939,14064,14065,14094,14095,14123],{},[344,13940,13942,13969],{"className":13941,"translate":348},[347],[344,13943,13945],{"className":13944},[352],[354,13946,13947],{"xmlns":356},[358,13948,13949,13967],{},[361,13950,13951,13957,13959,13965],{},[822,13952,13953,13955],{},[364,13954,2175],{},[364,13956,448],{},[450,13958,593],{"stretchy":592},[822,13960,13961,13963],{},[364,13962,815],{},[364,13964,1721],{},[450,13966,602],{"stretchy":592},[368,13968,4739],{"encoding":370},[344,13970,13972],{"className":13971,"ariaHidden":375},[374],[344,13973,13975,13978,14018,14021,14061],{"className":13974},[379],[344,13976],{"className":13977,"style":2281},[383],[344,13979,13981,13984],{"className":13980},[388],[344,13982,2175],{"className":13983,"style":2288},[388,389],[344,13985,13987],{"className":13986},[513],[344,13988,13990,14010],{"className":13989},[517,900],[344,13991,13993,14007],{"className":13992},[521],[344,13994,13996],{"className":13995,"style":2301},[525],[344,13997,13998,14001],{"style":2304},[344,13999],{"className":14000,"style":533},[532],[344,14002,14004],{"className":14003},[537,538,539,540],[344,14005,448],{"className":14006,"style":483},[388,389,540],[344,14008,924],{"className":14009},[923],[344,14011,14013],{"className":14012},[521],[344,14014,14016],{"className":14015,"style":2323},[525],[344,14017],{},[344,14019,593],{"className":14020},[668],[344,14022,14024,14027],{"className":14023},[388],[344,14025,815],{"className":14026},[388,389],[344,14028,14030],{"className":14029},[513],[344,14031,14033,14053],{"className":14032},[517,900],[344,14034,14036,14050],{"className":14035},[521],[344,14037,14039],{"className":14038,"style":1761},[525],[344,14040,14041,14044],{"style":910},[344,14042],{"className":14043,"style":533},[532],[344,14045,14047],{"className":14046},[537,538,539,540],[344,14048,1721],{"className":14049},[388,389,540],[344,14051,924],{"className":14052},[923],[344,14054,14056],{"className":14055},[521],[344,14057,14059],{"className":14058,"style":931},[525],[344,14060],{},[344,14062,602],{"className":14063},[701]," 只要有一个负特征值 ",[344,14066,14068,14082],{"className":14067,"translate":348},[347],[344,14069,14071],{"className":14070},[352],[354,14072,14073],{"xmlns":356},[358,14074,14075,14079],{},[361,14076,14077],{},[364,14078,4997],{},[368,14080,14081],{"encoding":370},"\\lambda",[344,14083,14085],{"className":14084,"ariaHidden":375},[374],[344,14086,14088,14091],{"className":14087},[379],[344,14089],{"className":14090,"style":3391},[383],[344,14092,4997],{"className":14093},[388,389],"，对应特征向量 ",[344,14096,14098,14111],{"className":14097,"translate":348},[347],[344,14099,14101],{"className":14100},[352],[354,14102,14103],{"xmlns":356},[358,14104,14105,14109],{},[361,14106,14107],{},[364,14108,4852],{},[368,14110,4852],{"encoding":370},[344,14112,14114],{"className":14113,"ariaHidden":375},[374],[344,14115,14117,14120],{"className":14116},[379],[344,14118],{"className":14119,"style":865},[383],[344,14121,4852],{"className":14122,"style":4886},[388,389],"，那么",[344,14125,14127],{"className":14126,"translate":348},[1047],[344,14128,14130,14172],{"className":14129,"translate":348},[347],[344,14131,14133],{"className":14132},[352],[354,14134,14135],{"xmlns":356,"display":1056},[358,14136,14137,14169],{},[361,14138,14139,14145,14151,14153,14159,14161,14163,14165,14167],{},[454,14140,14141,14143],{},[364,14142,4852],{},[364,14144,3547],{},[822,14146,14147,14149],{},[364,14148,2175],{},[364,14150,448],{},[450,14152,593],{"stretchy":592},[822,14154,14155,14157],{},[364,14156,815],{},[364,14158,1721],{},[450,14160,602],{"stretchy":592},[364,14162,4852],{},[450,14164,6876],{},[587,14166,1096],{},[450,14168,831],{"separator":375},[368,14170,14171],{"encoding":370},"v^T H_f(x_c) v \u003C 0,",[344,14173,14175,14308],{"className":14174,"ariaHidden":375},[374],[344,14176,14178,14181,14210,14250,14253,14293,14296,14299,14302,14305],{"className":14177},[379],[344,14179],{"className":14180,"style":5465},[383],[344,14182,14184,14187],{"className":14183},[388],[344,14185,4852],{"className":14186,"style":4886},[388,389],[344,14188,14190],{"className":14189},[513],[344,14191,14193],{"className":14192},[517],[344,14194,14196],{"className":14195},[521],[344,14197,14199],{"className":14198,"style":1661},[525],[344,14200,14201,14204],{"style":1682},[344,14202],{"className":14203,"style":533},[532],[344,14205,14207],{"className":14206},[537,538,539,540],[344,14208,3547],{"className":14209,"style":3849},[388,389,540],[344,14211,14213,14216],{"className":14212},[388],[344,14214,2175],{"className":14215,"style":2288},[388,389],[344,14217,14219],{"className":14218},[513],[344,14220,14222,14242],{"className":14221},[517,900],[344,14223,14225,14239],{"className":14224},[521],[344,14226,14228],{"className":14227,"style":2301},[525],[344,14229,14230,14233],{"style":2304},[344,14231],{"className":14232,"style":533},[532],[344,14234,14236],{"className":14235},[537,538,539,540],[344,14237,448],{"className":14238,"style":483},[388,389,540],[344,14240,924],{"className":14241},[923],[344,14243,14245],{"className":14244},[521],[344,14246,14248],{"className":14247,"style":2323},[525],[344,14249],{},[344,14251,593],{"className":14252},[668],[344,14254,14256,14259],{"className":14255},[388],[344,14257,815],{"className":14258},[388,389],[344,14260,14262],{"className":14261},[513],[344,14263,14265,14285],{"className":14264},[517,900],[344,14266,14268,14282],{"className":14267},[521],[344,14269,14271],{"className":14270,"style":1761},[525],[344,14272,14273,14276],{"style":910},[344,14274],{"className":14275,"style":533},[532],[344,14277,14279],{"className":14278},[537,538,539,540],[344,14280,1721],{"className":14281},[388,389,540],[344,14283,924],{"className":14284},[923],[344,14286,14288],{"className":14287},[521],[344,14289,14291],{"className":14290,"style":931},[525],[344,14292],{},[344,14294,602],{"className":14295},[701],[344,14297,4852],{"className":14298,"style":4886},[388,389],[344,14300],{"className":14301,"style":488},[487],[344,14303,6876],{"className":14304},[492],[344,14306],{"className":14307,"style":488},[487],[344,14309,14311,14314,14317],{"className":14310},[379],[344,14312],{"className":14313,"style":1943},[383],[344,14315,1096],{"className":14316},[388],[344,14318,831],{"className":14319},[937],[15,14321,14322,14323,14351,14352,14380],{},"沿 ",[344,14324,14326,14339],{"className":14325,"translate":348},[347],[344,14327,14329],{"className":14328},[352],[354,14330,14331],{"xmlns":356},[358,14332,14333,14337],{},[361,14334,14335],{},[364,14336,4852],{},[368,14338,4852],{"encoding":370},[344,14340,14342],{"className":14341,"ariaHidden":375},[374],[344,14343,14345,14348],{"className":14344},[379],[344,14346],{"className":14347,"style":865},[383],[344,14349,4852],{"className":14350,"style":4886},[388,389]," 移动一小段距离会使 ",[344,14353,14355,14368],{"className":14354,"translate":348},[347],[344,14356,14358],{"className":14357},[352],[354,14359,14360],{"xmlns":356},[358,14361,14362,14366],{},[361,14363,14364],{},[364,14365,448],{},[368,14367,448],{"encoding":370},[344,14369,14371],{"className":14370,"ariaHidden":375},[374],[344,14372,14374,14377],{"className":14373},[379],[344,14375],{"className":14376,"style":479},[383],[344,14378,448],{"className":14379,"style":483},[388,389]," 在二阶意义上减小：",[344,14382,14384],{"className":14383,"translate":348},[1047],[344,14385,14387,14476],{"className":14386,"translate":348},[347],[344,14388,14390],{"className":14389},[352],[354,14391,14392],{"xmlns":356,"display":1056},[358,14393,14394,14473],{},[361,14395,14396,14398,14400,14406,14408,14411,14413,14415,14417,14419,14421,14427,14429,14431,14437,14443,14445,14451,14457,14459,14465,14467,14469,14471],{},[364,14397,448],{},[450,14399,593],{"stretchy":592},[822,14401,14402,14404],{},[364,14403,815],{},[364,14405,1721],{},[450,14407,3508],{},[364,14409,14410],{},"t",[364,14412,4852],{},[450,14414,602],{"stretchy":592},[450,14416,3372],{},[364,14418,448],{},[450,14420,593],{"stretchy":592},[822,14422,14423,14425],{},[364,14424,815],{},[364,14426,1721],{},[450,14428,602],{"stretchy":592},[450,14430,13563],{},[1102,14432,14433,14435],{},[587,14434,828],{},[587,14436,589],{},[454,14438,14439,14441],{},[364,14440,14410],{},[587,14442,589],{},[3609,14444,12986],{},[454,14446,14447,14449],{},[364,14448,4852],{},[364,14450,3547],{},[822,14452,14453,14455],{},[364,14454,2175],{},[364,14456,448],{},[450,14458,593],{"stretchy":592},[822,14460,14461,14463],{},[364,14462,815],{},[364,14464,1721],{},[450,14466,602],{"stretchy":592},[364,14468,4852],{},[450,14470,6876],{},[587,14472,7260],{},[368,14474,14475],{"encoding":370},"f(x_c+tv) - f(x_c) \\approx \\frac{1}{2}t^2\\, v^T H_f(x_c) v \u003C 0.",[344,14477,14479,14540,14564,14628,14855],{"className":14478,"ariaHidden":375},[374],[344,14480,14482,14485,14488,14491,14531,14534,14537],{"className":14481},[379],[344,14483],{"className":14484,"style":884},[383],[344,14486,448],{"className":14487,"style":483},[388,389],[344,14489,593],{"className":14490},[668],[344,14492,14494,14497],{"className":14493},[388],[344,14495,815],{"className":14496},[388,389],[344,14498,14500],{"className":14499},[513],[344,14501,14503,14523],{"className":14502},[517,900],[344,14504,14506,14520],{"className":14505},[521],[344,14507,14509],{"className":14508,"style":1761},[525],[344,14510,14511,14514],{"style":910},[344,14512],{"className":14513,"style":533},[532],[344,14515,14517],{"className":14516},[537,538,539,540],[344,14518,1721],{"className":14519},[388,389,540],[344,14521,924],{"className":14522},[923],[344,14524,14526],{"className":14525},[521],[344,14527,14529],{"className":14528,"style":931},[525],[344,14530],{},[344,14532],{"className":14533,"style":2138},[487],[344,14535,3508],{"className":14536},[2142],[344,14538],{"className":14539,"style":2138},[487],[344,14541,14543,14546,14549,14552,14555,14558,14561],{"className":14542},[379],[344,14544],{"className":14545,"style":884},[383],[344,14547,14410],{"className":14548},[388,389],[344,14550,4852],{"className":14551,"style":4886},[388,389],[344,14553,602],{"className":14554},[701],[344,14556],{"className":14557,"style":2138},[487],[344,14559,3372],{"className":14560},[2142],[344,14562],{"className":14563,"style":2138},[487],[344,14565,14567,14570,14573,14576,14616,14619,14622,14625],{"className":14566},[379],[344,14568],{"className":14569,"style":884},[383],[344,14571,448],{"className":14572,"style":483},[388,389],[344,14574,593],{"className":14575},[668],[344,14577,14579,14582],{"className":14578},[388],[344,14580,815],{"className":14581},[388,389],[344,14583,14585],{"className":14584},[513],[344,14586,14588,14608],{"className":14587},[517,900],[344,14589,14591,14605],{"className":14590},[521],[344,14592,14594],{"className":14593,"style":1761},[525],[344,14595,14596,14599],{"style":910},[344,14597],{"className":14598,"style":533},[532],[344,14600,14602],{"className":14601},[537,538,539,540],[344,14603,1721],{"className":14604},[388,389,540],[344,14606,924],{"className":14607},[923],[344,14609,14611],{"className":14610},[521],[344,14612,14614],{"className":14613,"style":931},[525],[344,14615],{},[344,14617,602],{"className":14618},[701],[344,14620],{"className":14621,"style":488},[487],[344,14623,13563],{"className":14624},[492],[344,14626],{"className":14627,"style":488},[487],[344,14629,14631,14634,14696,14725,14728,14757,14797,14800,14840,14843,14846,14849,14852],{"className":14630},[379],[344,14632],{"className":14633,"style":3868},[383],[344,14635,14637,14640,14693],{"className":14636},[388],[344,14638],{"className":14639},[668,1332],[344,14641,14643],{"className":14642},[1102],[344,14644,14646,14685],{"className":14645},[517,900],[344,14647,14649,14682],{"className":14648},[521],[344,14650,14652,14663,14671],{"className":14651,"style":3887},[525],[344,14653,14654,14657],{"style":1348},[344,14655],{"className":14656,"style":1352},[532],[344,14658,14660],{"className":14659},[388],[344,14661,589],{"className":14662},[388],[344,14664,14665,14668],{"style":1402},[344,14666],{"className":14667,"style":1352},[532],[344,14669],{"className":14670,"style":1410},[1409],[344,14672,14673,14676],{"style":1413},[344,14674],{"className":14675,"style":1352},[532],[344,14677,14679],{"className":14678},[388],[344,14680,828],{"className":14681},[388],[344,14683,924],{"className":14684},[923],[344,14686,14688],{"className":14687},[521],[344,14689,14691],{"className":14690,"style":3927},[525],[344,14692],{},[344,14694],{"className":14695},[701,1332],[344,14697,14699,14702],{"className":14698},[388],[344,14700,14410],{"className":14701},[388,389],[344,14703,14705],{"className":14704},[513],[344,14706,14708],{"className":14707},[517],[344,14709,14711],{"className":14710},[521],[344,14712,14714],{"className":14713,"style":2369},[525],[344,14715,14716,14719],{"style":1682},[344,14717],{"className":14718,"style":533},[532],[344,14720,14722],{"className":14721},[537,538,539,540],[344,14723,589],{"className":14724},[388,540],[344,14726],{"className":14727,"style":941},[487],[344,14729,14731,14734],{"className":14730},[388],[344,14732,4852],{"className":14733,"style":4886},[388,389],[344,14735,14737],{"className":14736},[513],[344,14738,14740],{"className":14739},[517],[344,14741,14743],{"className":14742},[521],[344,14744,14746],{"className":14745,"style":1661},[525],[344,14747,14748,14751],{"style":1682},[344,14749],{"className":14750,"style":533},[532],[344,14752,14754],{"className":14753},[537,538,539,540],[344,14755,3547],{"className":14756,"style":3849},[388,389,540],[344,14758,14760,14763],{"className":14759},[388],[344,14761,2175],{"className":14762,"style":2288},[388,389],[344,14764,14766],{"className":14765},[513],[344,14767,1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Hessian 有很多机会获得这样的负方向。只有当这些方向都不向下弯曲时，非退化局部极小值才能存活。",[11,14871,14873],{"id":14872},"kacrice-公式","Kac–Rice 公式",[15,14875,14876,14877,702],{},"随机矩阵论证解释了为什么随机 Hessian 不太可能正定。要把这一论证与实际随机函数联系起来，需要一种计数临界点并按指数分类的方法。标准工具是 ",[324,14878,14873],{},[15,14880,433,14881,14909,14910,14994,14995,15073],{},[344,14882,14884,14897],{"className":14883,"translate":348},[347],[344,14885,14887],{"className":14886},[352],[354,14888,14889],{"xmlns":356},[358,14890,14891,14895],{},[361,14892,14893],{},[364,14894,448],{},[368,14896,448],{"encoding":370},[344,14898,14900],{"className":14899,"ariaHidden":375},[374],[344,14901,14903,14906],{"className":14902},[379],[344,14904],{"className":14905,"style":479},[383],[344,14907,448],{"className":14908,"style":483},[388,389]," 是区域 ",[344,14911,14913,14937],{"className":14912,"translate":348},[347],[344,14914,14916],{"className":14915},[352],[354,14917,14918],{"xmlns":356},[358,14919,14920,14934],{},[361,14921,14922,14925,14928],{},[364,14923,14924],{},"D",[450,14926,14927],{},"⊂",[454,14929,14930,14932],{},[364,14931,459],{"mathvariant":458},[364,14933,366],{},[368,14935,14936],{"encoding":370},"D\\subset\\mathbb{R}^N",[344,14938,14940,14959],{"className":14939,"ariaHidden":375},[374],[344,14941,14943,14947,14950,14953,14956],{"className":14942},[379],[344,14944],{"className":14945,"style":14946},[383],"height:0.7224em;vertical-align:-0.0391em;",[344,14948,14924],{"className":14949,"style":12019},[388,389],[344,14951],{"className":14952,"style":488},[487],[344,14954,14927],{"className":14955},[492],[344,14957],{"className":14958,"style":488},[487],[344,14960,14962,14965],{"className":14961},[379],[344,14963],{"className":14964,"style":502},[383],[344,14966,14968,14971],{"className":14967},[388],[344,14969,459],{"className":14970},[388,509],[344,14972,14974],{"className":14973},[513],[344,14975,14977],{"className":14976},[517],[344,14978,14980],{"className":14979},[521],[344,14981,14983],{"className":14982,"style":502},[525],[344,14984,14985,14988],{"style":528},[344,14986],{"className":14987,"style":533},[532],[344,14989,14991],{"className":14990},[537,538,539,540],[344,14992,366],{"className":14993,"style":390},[388,389,540]," 上足够正则的随机场。对 ",[344,14996,14998,15022],{"className":14997,"translate":348},[347],[344,14999,15001],{"className":15000},[352],[354,15002,15003],{"xmlns":356},[358,15004,15005,15019],{},[361,15006,15007,15009,15012,15015,15017],{},[587,15008,1096],{},[450,15010,15011],{},"≤",[364,15013,15014],{},"k",[450,15016,15011],{},[364,15018,366],{},[368,15020,15021],{"encoding":370},"0\\le k\\le N",[344,15023,15025,15044,15064],{"className":15024,"ariaHidden":375},[374],[344,15026,15028,15032,15035,15038,15041],{"className":15027},[379],[344,15029],{"className":15030,"style":15031},[383],"height:0.7804em;vertical-align:-0.136em;",[344,15033,1096],{"className":15034},[388],[344,15036],{"className":15037,"style":488},[487],[344,15039,15011],{"className":15040},[492],[344,15042],{"className":15043,"style":488},[487],[344,15045,15047,15051,15055,15058,15061],{"className":15046},[379],[344,15048],{"className":15049,"style":15050},[383],"height:0.8304em;vertical-align:-0.136em;",[344,15052,15014],{"className":15053,"style":15054},[388,389],"margin-right:0.0315em;",[344,15056],{"className":15057,"style":488},[487],[344,15059,15011],{"className":15060},[492],[344,15062],{"className":15063,"style":488},[487],[344,15065,15067,15070],{"className":15066},[379],[344,15068],{"className":15069,"style":384},[383],[344,15071,366],{"className":15072,"style":390},[388,389],"，定义",[344,15075,15077],{"className":15076,"translate":348},[1047],[344,15078,15080,15166],{"className":15079,"translate":348},[347],[344,15081,15083],{"className":15082},[352],[354,15084,15085],{"xmlns":356,"display":1056},[358,15086,15087,15163],{},[361,15088,15089,15095,15097,15099,15101,15103,15105,15107,15109,15111,15113,15115,15117,15119,15121,15123,15125,15127,15129,15131,15133,15135,15137,15139,15141,15147,15149,15151,15153,15155,15157,15159,15161],{},[822,15090,15091,15093],{},[364,15092,585],{},[364,15094,15014],{},[450,15096,593],{"stretchy":592},[364,15098,14924],{},[450,15100,602],{"stretchy":592},[450,15102,452],{},[450,15104,818],{},[364,15106,6860],{"mathvariant":1063},[450,15108,6863],{"stretchy":592},[364,15110,815],{},[450,15112,580],{},[364,15114,14924],{},[450,15116,452],{},[364,15118,1064],{"mathvariant":1063},[364,15120,448],{},[450,15122,593],{"stretchy":592},[364,15124,815],{},[450,15126,602],{"stretchy":592},[450,15128,818],{},[587,15130,1096],{},[450,15132,831],{"separator":375},[3609,15134,7229],{},[364,15136,6840],{"mathvariant":1063},[450,15138,6843],{},[450,15140,593],{"stretchy":592},[822,15142,15143,15145],{},[364,15144,2175],{},[364,15146,448],{},[450,15148,593],{"stretchy":592},[364,15150,815],{},[450,15152,602],{"stretchy":592},[450,15154,602],{"stretchy":592},[450,15156,818],{},[364,15158,15014],{},[450,15160,6881],{"stretchy":592},[450,15162,831],{"separator":375},[368,15164,15165],{"encoding":370},"C_k(D) := \\#\\{x\\in D : \\nabla f(x)=0,\\ \\operatorname{index}(H_f(x))=k\\},",[344,15167,15169,15234,15258,15276,15306,15392],{"className":15168,"ariaHidden":375},[374],[344,15170,15172,15175,15216,15219,15222,15225,15228,15231],{"className":15171},[379],[344,15173],{"className":15174,"style":884},[383],[344,15176,15178,15181],{"className":15177},[388],[344,15179,585],{"className":15180,"style":640},[388,389],[344,15182,15184],{"className":15183},[513],[344,15185,15187,15208],{"className":15186},[517,900],[344,15188,15190,15205],{"className":15189},[521],[344,15191,15193],{"className":15192,"style":2301},[525],[344,15194,15196,15199],{"style":15195},"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;",[344,15197],{"className":15198,"style":533},[532],[344,15200,15202],{"className":15201},[537,538,539,540],[344,15203,15014],{"className":15204,"style":15054},[388,389,540],[344,15206,924],{"className":15207},[923],[344,15209,15211],{"className":15210},[521],[344,15212,15214],{"className":15213,"style":931},[525],[344,15215],{},[344,15217,593],{"className":15218},[668],[344,15220,14924],{"className":15221,"style":12019},[388,389],[344,15223,602],{"className":15224},[701],[344,15226],{"className":15227,"style":488},[487],[344,15229,1226],{"className":15230},[492],[344,15232],{"className":15233,"style":488},[487],[344,15235,15237,15240,15243,15246,15249,15252,15255],{"className":15236},[379],[344,15238],{"className":15239,"style":884},[383],[344,15241,6860],{"className":15242},[388],[344,15244,6863],{"className":15245},[668],[344,15247,815],{"className":15248},[388,389],[344,15250],{"className":15251,"style":488},[487],[344,15253,580],{"className":15254},[492],[344,15256],{"className":15257,"style":488},[487],[344,15259,15261,15264,15267,15270,15273],{"className":15260},[379],[344,15262],{"className":15263,"style":384},[383],[344,15265,14924],{"className":15266,"style":12019},[388,389],[344,15268],{"className":15269,"style":488},[487],[344,15271,452],{"className":15272},[492],[344,15274],{"className":15275,"style":488},[487],[344,15277,15279,15282,15285,15288,15291,15294,15297,15300,15303],{"className":15278},[379],[344,15280],{"className":15281,"style":884},[383],[344,15283,1064],{"className":15284},[388],[344,15286,448],{"className":15287,"style":483},[388,389],[344,15289,593],{"className":15290},[668],[344,15292,815],{"className":15293},[388,389],[344,15295,602],{"className":15296},[701],[344,15298],{"className":15299,"style":488},[487],[344,15301,818],{"className":15302},[492],[344,15304],{"className":15305,"style":488},[487],[344,15307,15309,15312,15315,15318,15321,15324,15330,15333,15373,15376,15379,15383,15386,15389],{"className":15308},[379],[344,15310],{"className":15311,"style":2281},[383],[344,15313,1096],{"className":15314},[388],[344,15316,831],{"className":15317},[937],[344,15319,7229],{"className":15320},[487],[344,15322],{"className":15323,"style":941},[487],[344,15325,15327],{"className":15326},[5207],[344,15328,6840],{"className":15329},[388,6902],[344,15331,593],{"className":15332},[668],[344,15334,15336,15339],{"className":15335},[388],[344,15337,2175],{"className":15338,"style":2288},[388,389],[344,15340,15342],{"className":15341},[513],[344,15343,15345,15365],{"className":15344},[517,900],[344,15346,15348,15362],{"className":15347},[521],[344,15349,15351],{"className":15350,"style":2301},[525],[344,15352,15353,15356],{"style":2304},[344,15354],{"className":15355,"style":533},[532],[344,15357,15359],{"className":15358},[537,538,539,540],[344,15360,448],{"className":15361,"style":483},[388,389,540],[344,15363,924],{"className":15364},[923],[344,15366,15368],{"className":15367},[521],[344,15369,15371],{"className":15370,"style":2323},[525],[344,15372],{},[344,15374,593],{"className":15375},[668],[344,15377,815],{"className":15378},[388,389],[344,15380,15382],{"className":15381},[701],"))",[344,15384],{"className":15385,"style":488},[487],[344,15387,818],{"className":15388},[492],[344,15390],{"className":15391,"style":488},[487],[344,15393,15395,15398,15401,15404],{"className":15394},[379],[344,15396],{"className":15397,"style":884},[383],[344,15399,15014],{"className":15400,"style":15054},[388,389],[344,15402,6881],{"className":15403},[701],[344,15405,831],{"className":15406},[937],[15,15408,15409,15410,15438,15439,15467],{},"即 ",[344,15411,15413,15426],{"className":15412,"translate":348},[347],[344,15414,15416],{"className":15415},[352],[354,15417,15418],{"xmlns":356},[358,15419,15420,15424],{},[361,15421,15422],{},[364,15423,14924],{},[368,15425,14924],{"encoding":370},[344,15427,15429],{"className":15428,"ariaHidden":375},[374],[344,15430,15432,15435],{"className":15431},[379],[344,15433],{"className":15434,"style":384},[383],[344,15436,14924],{"className":15437,"style":12019},[388,389]," 中指数为 ",[344,15440,15442,15455],{"className":15441,"translate":348},[347],[344,15443,15445],{"className":15444},[352],[354,15446,15447],{"xmlns":356},[358,15448,15449,15453],{},[361,15450,15451],{},[364,15452,15014],{},[368,15454,15014],{"encoding":370},[344,15456,15458],{"className":15457,"ariaHidden":375},[374],[344,15459,15461,15464],{"className":15460},[379],[344,15462],{"className":15463,"style":3391},[383],[344,15465,15014],{"className":15466,"style":15054},[388,389]," 的临界点个数。在正则性与非退化性假设下，Kac–Rice 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f(x)}(0)",[344,16169,16171],{"className":16170,"ariaHidden":375},[374],[344,16172,16174,16178,16233,16236,16239],{"className":16173},[379],[344,16175],{"className":16176,"style":16177},[383],"height:1.1052em;vertical-align:-0.3552em;",[344,16179,16181,16184],{"className":16180},[388],[344,16182,15],{"className":16183},[388,389],[344,16185,16187],{"className":16186},[513],[344,16188,16190,16225],{"className":16189},[517,900],[344,16191,16193,16222],{"className":16192},[521],[344,16194,16196],{"className":16195,"style":16046},[525],[344,16197,16198,16201],{"style":16049},[344,16199],{"className":16200,"style":533},[532],[344,16202,16204],{"className":16203},[537,538,539,540],[344,16205,16207,16210,16213,16216,16219],{"className":16206},[388,540],[344,16208,1064],{"className":16209},[388,540],[344,16211,448],{"className":16212,"style":483},[388,389,540],[344,16214,593],{"className":16215},[668,540],[344,16217,815],{"className":16218},[388,389,540],[344,16220,602],{"className":16221},[701,540],[344,16223,924],{"className":16224},[923],[344,16226,16228],{"className":16227},[521],[344,16229,16231],{"className":16230,"style":16083},[525],[344,16232],{},[344,16234,593],{"className":16235},[668],[344,16237,1096],{"className":16238},[388],[344,16240,602],{"className":16241},[701]," 是梯度在零处的密度。行列式因子的出现，是因为梯度映射的零点要按局部雅可比体积变化来计数。指示函数则选出所需的指数。",[15,16244,16245,16246,8905],{},"方程 (3) 使问题变得精确。计数非退化局部极小值就是取 ",[344,16247,16249,16267],{"className":16248,"translate":348},[347],[344,16250,16252],{"className":16251},[352],[354,16253,16254],{"xmlns":356},[358,16255,16256,16264],{},[361,16257,16258,16260,16262],{},[364,16259,15014],{},[450,16261,818],{},[587,16263,1096],{},[368,16265,16266],{"encoding":370},"k=0",[344,16268,16270,16288],{"className":16269,"ariaHidden":375},[374],[344,16271,16273,16276,16279,16282,16285],{"className":16272},[379],[344,16274],{"className":16275,"style":3391},[383],[344,16277,15014],{"className":16278,"style":15054},[388,389],[344,16280],{"className":16281,"style":488},[487],[344,16283,818],{"className":16284},[492],[344,16286],{"className":16287,"style":488},[487],[344,16289,16291,16294],{"className":16290},[379],[344,16292],{"className":16293,"style":7133},[383],[344,16295,1096],{"className":16296},[388],[344,16298,16300],{"className":16299,"translate":348},[1047],[344,16301,16303,16380],{"className":16302,"translate":348},[347],[344,16304,16306],{"className":16305},[352],[354,16307,16308],{"xmlns":356,"display":1056},[358,16309,16310,16377],{},[361,16311,16312,16318,16320,16322,16324,16326,16328,16330,16332,16334,16336,16338,16340,16342,16344,16346,16348,16350,16352,16354,16356,16362,16364,16366,16368,16371,16373,16375],{},[822,16313,16314,16316],{},[364,16315,585],{},[587,16317,1096],{},[450,16319,593],{"stretchy":592},[364,16321,14924],{},[450,16323,602],{"stretchy":592},[450,16325,818],{},[364,16327,6860],{"mathvariant":1063},[450,16329,6863],{"stretchy":592},[364,16331,815],{},[450,16333,580],{},[364,16335,14924],{},[450,16337,452],{},[364,16339,1064],{"mathvariant":1063},[364,16341,448],{},[450,16343,593],{"stretchy":592},[364,16345,815],{},[450,16347,602],{"stretchy":592},[450,16349,818],{},[587,16351,1096],{},[450,16353,831],{"separator":375},[3609,16355,7229],{},[822,16357,16358,16360],{},[364,16359,2175],{},[364,16361,448],{},[450,16363,593],{"stretchy":592},[364,16365,815],{},[450,16367,602],{"stretchy":592},[450,16369,16370],{},"≻",[587,16372,1096],{},[450,16374,6881],{"stretchy":592},[364,16376,1192],{"mathvariant":1063},[368,16378,16379],{"encoding":370},"C_0(D) = \\#\\{x\\in D : \\nabla f(x)=0,\\ H_f(x)\\succ 0\\}.",[344,16381,16383,16447,16471,16489,16519,16595],{"className":16382,"ariaHidden":375},[374],[344,16384,16386,16389,16429,16432,16435,16438,16441,16444],{"className":16385},[379],[344,16387],{"className":16388,"style":884},[383],[344,16390,16392,16395],{"className":16391},[388],[344,16393,585],{"className":16394,"style":640},[388,389],[344,16396,16398],{"className":16397},[513],[344,16399,16401,16421],{"className":16400},[517,900],[344,16402,16404,16418],{"className":16403},[521],[344,16405,16407],{"className":16406,"style":907},[525],[344,16408,16409,16412],{"style":15195},[344,16410],{"className":16411,"style":533},[532],[344,16413,16415],{"className":16414},[537,538,539,540],[344,16416,1096],{"className":16417},[388,540],[344,16419,924],{"className":16420},[923],[344,16422,16424],{"className":16423},[521],[344,16425,16427],{"className":16426,"style":931},[525],[344,16428],{},[344,16430,593],{"className":16431},[668],[344,16433,14924],{"className":16434,"style":12019},[388,389],[344,16436,602],{"className":16437},[701],[344,16439],{"className":16440,"style":488},[487],[344,16442,818],{"className":16443},[492],[344,16445],{"className":16446,"style":488},[487],[344,16448,16450,16453,16456,16459,16462,16465,16468],{"className":16449},[379],[344,16451],{"className":16452,"style":884},[383],[344,16454,6860],{"className":16455},[388],[344,16457,6863],{"className":16458},[668],[344,16460,815],{"className":16461},[388,389],[344,16463],{"className":16464,"style":488},[487],[344,16466,580],{"className":16467},[492],[344,16469],{"className":16470,"style":488},[487],[344,16472,16474,16477,16480,16483,16486],{"className":16473},[379],[344,16475],{"className":16476,"style":384},[383],[344,16478,14924],{"className":16479,"style":12019},[388,389],[344,16481],{"className":16482,"style":488},[487],[344,16484,452],{"className":16485},[492],[344,16487],{"className":16488,"style":488},[487],[344,16490,16492,16495,16498,16501,16504,16507,16510,16513,16516],{"className":16491},[379],[344,16493],{"className":16494,"style":884},[383],[344,16496,1064],{"className":16497},[388],[344,16499,448],{"className":16500,"style":483},[388,389],[344,16502,593],{"className":16503},[668],[344,16505,815],{"className":16506},[388,389],[344,16508,602],{"className":16509},[701],[344,16511],{"className":16512,"style":488},[487],[344,16514,818],{"className":16515},[492],[344,16517],{"className":16518,"style":488},[487],[344,16520,16522,16525,16528,16531,16534,16537,16577,16580,16583,16586,16589,16592],{"className":16521},[379],[344,16523],{"className":16524,"style":2281},[383],[344,16526,1096],{"className":16527},[388],[344,16529,831],{"className":16530},[937],[344,16532,7229],{"className":16533},[487],[344,16535],{"className":16536,"style":941},[487],[344,16538,16540,16543],{"className":16539},[388],[344,16541,2175],{"className":16542,"style":2288},[388,389],[344,16544,16546],{"className":16545},[513],[344,16547,16549,16569],{"className":16548},[517,900],[344,16550,16552,16566],{"className":16551},[521],[344,16553,16555],{"className":16554,"style":2301},[525],[344,16556,16557,16560],{"style":2304},[344,16558],{"className":16559,"style":533},[532],[344,16561,16563],{"className":16562},[537,538,539,540],[344,16564,448],{"className":16565,"style":483},[388,389,540],[344,16567,924],{"className":16568},[923],[344,16570,16572],{"className":16571},[521],[344,16573,16575],{"className":16574,"style":2323},[525],[344,16576],{},[344,16578,593],{"className":16579},[668],[344,16581,815],{"className":16582},[388,389],[344,16584,602],{"className":16585},[701],[344,16587],{"className":16588,"style":488},[487],[344,16590,16370],{"className":16591},[492],[344,16593],{"className":16594,"style":488},[487],[344,16596,16598,16601,16604,16607],{"className":16597},[379],[344,16599],{"className":16600,"style":884},[383],[344,16602,1096],{"className":16603},[388],[344,16605,6881],{"className":16606},[701],[344,16608,1192],{"className":16609},[388],[15,16611,16612,16613,16708],{},"计数鞍点则是对 ",[344,16614,16616,16642],{"className":16615,"translate":348},[347],[344,16617,16619],{"className":16618},[352],[354,16620,16621],{"xmlns":356},[358,16622,16623,16639],{},[361,16624,16625,16627,16629,16631,16633,16635,16637],{},[587,16626,828],{},[450,16628,15011],{},[364,16630,15014],{},[450,16632,15011],{},[364,16634,366],{},[450,16636,3372],{},[587,16638,828],{},[368,16640,16641],{"encoding":370},"1\\le k\\le N-1",[344,16643,16645,16663,16681,16699],{"className":16644,"ariaHidden":375},[374],[344,16646,16648,16651,16654,16657,16660],{"className":16647},[379],[344,16649],{"className":16650,"style":15031},[383],[344,16652,828],{"className":16653},[388],[344,16655],{"className":16656,"style":488},[487],[344,16658,15011],{"className":16659},[492],[344,16661],{"className":16662,"style":488},[487],[344,16664,16666,16669,16672,16675,16678],{"className":16665},[379],[344,16667],{"className":16668,"style":15050},[383],[344,16670,15014],{"className":16671,"style":15054},[388,389],[344,16673],{"className":16674,"style":488},[487],[344,16676,15011],{"className":16677},[492],[344,16679],{"className":16680,"style":488},[487],[344,16682,16684,16687,16690,16693,16696],{"className":16683},[379],[344,16685],{"className":16686,"style":2131},[383],[344,16688,366],{"className":16689,"style":390},[388,389],[344,16691],{"className":16692,"style":2138},[487],[344,16694,3372],{"className":16695},[2142],[344,16697],{"className":16698,"style":2138},[487],[344,16700,16702,16705],{"className":16701},[379],[344,16703],{"className":16704,"style":7133},[383],[344,16706,828],{"className":16707},[388]," 求和。",[15,16710,16711,16712,16783],{},"对许多高斯随机场，在 ",[344,16713,16715,16741],{"className":16714,"translate":348},[347],[344,16716,16718],{"className":16717},[352],[354,16719,16720],{"xmlns":356},[358,16721,16722,16738],{},[361,16723,16724,16726,16728,16730,16732,16734,16736],{},[364,16725,1064],{"mathvariant":1063},[364,16727,448],{},[450,16729,593],{"stretchy":592},[364,16731,815],{},[450,16733,602],{"stretchy":592},[450,16735,818],{},[587,16737,1096],{},[368,16739,16740],{"encoding":370},"\\nabla f(x)=0",[344,16742,16744,16774],{"className":16743,"ariaHidden":375},[374],[344,16745,16747,16750,16753,16756,16759,16762,16765,16768,16771],{"className":16746},[379],[344,16748],{"className":16749,"style":884},[383],[344,16751,1064],{"className":16752},[388],[344,16754,448],{"className":16755,"style":483},[388,389],[344,16757,593],{"className":16758},[668],[344,16760,815],{"className":16761},[388,389],[344,16763,602],{"className":16764},[701],[344,16766],{"className":16767,"style":488},[487],[344,16769,818],{"className":16770},[492],[344,16772],{"className":16773,"style":488},[487],[344,16775,16777,16780],{"className":16776},[379],[344,16778],{"className":16779,"style":7133},[383],[344,16781,1096],{"className":16782},[388]," 条件下，Hessian 的条件分布是一个正交不变的高斯矩阵，通常是 GOE 型矩阵加上一个依赖于场值的标量平移——这正是随机矩阵理论严格介入之处，Hessian 的指数分布控制着每一类临界点的期望个数。",[15,16785,16786,16787,16815],{},"在自旋玻璃模型中，这一方案已被详细执行。Bray 与 Dean 计算了大维空间上高斯场的平均临界点个数，把它作为能量与指数的函数。Aufinger、Ben Arous 与 Černý 利用 Kac–Rice 与 GOE 特征值渐近，对球面 ",[344,16788,16790,16803],{"className":16789,"translate":348},[347],[344,16791,16793],{"className":16792},[352],[354,16794,16795],{"xmlns":356},[358,16796,16797,16801],{},[361,16798,16799],{},[364,16800,15],{},[368,16802,15],{"encoding":370},[344,16804,16806],{"className":16805,"ariaHidden":375},[374],[344,16807,16809,16812],{"className":16808},[379],[344,16810],{"className":16811,"style":4879},[383],[344,16813,15],{"className":16814},[388,389],"-自旋自旋玻璃哈密顿量的复杂度给出了严格的渐近分析。这些工作展示了一种典型模式：临界点可能是指数级众多的，但主导指数通常不是 0——相反，在景观的大部分区域，占主导的临界点具有正比例的负特征值，即它们是鞍点。",[15,16817,16818],{},"因此，严格的概率陈述不是「每个高维函数都只有很少的局部极小值」，而是更接近：",[764,16820,16821],{},[15,16822,16823],{},"在广泛的高维随机景观模型中，期望的临界点计数以鞍点为主，而指数为零的临界点只构成一个趋于消失或指数级更小的比例，除非处于特殊的低能区域。",[11,16825,16827],{"id":16826},"morse-理论","Morse 理论",[15,16829,16830],{},"Morse 理论为指数提供了拓扑诠释。它本身并不证明极小值是罕见的，而是解释为什么指数是应当追踪的正确不变量。",[15,16832,433,16833,16907,16908,16936,16937,16965],{},[344,16834,16836,16859],{"className":16835,"translate":348},[347],[344,16837,16839],{"className":16838},[352],[354,16840,16841],{"xmlns":356},[358,16842,16843,16856],{},[361,16844,16845,16847,16849,16852,16854],{},[364,16846,448],{},[450,16848,452],{},[364,16850,16851],{},"M",[450,16853,464],{},[364,16855,459],{"mathvariant":458},[368,16857,16858],{"encoding":370},"f: M\\to\\mathbb{R}",[344,16860,16862,16880,16898],{"className":16861,"ariaHidden":375},[374],[344,16863,16865,16868,16871,16874,16877],{"className":16864},[379],[344,16866],{"className":16867,"style":479},[383],[344,16869,448],{"className":16870,"style":483},[388,389],[344,16872],{"className":16873,"style":488},[487],[344,16875,452],{"className":16876},[492],[344,16878],{"className":16879,"style":488},[487],[344,16881,16883,16886,16889,16892,16895],{"className":16882},[379],[344,16884],{"className":16885,"style":384},[383],[344,16887,16851],{"className":16888,"style":390},[388,389],[344,16890],{"className":16891,"style":488},[487],[344,16893,464],{"className":16894},[492],[344,16896],{"className":16897,"style":488},[487],[344,16899,16901,16904],{"className":16900},[379],[344,16902],{"className":16903,"style":559},[383],[344,16905,459],{"className":16906},[388,509]," 是紧光滑流形 ",[344,16909,16911,16924],{"className":16910,"translate":348},[347],[344,16912,16914],{"className":16913},[352],[354,16915,16916],{"xmlns":356},[358,16917,16918,16922],{},[361,16919,16920],{},[364,16921,16851],{},[368,16923,16851],{"encoding":370},[344,16925,16927],{"className":16926,"ariaHidden":375},[374],[344,16928,16930,16933],{"className":16929},[379],[344,16931],{"className":16932,"style":384},[383],[344,16934,16851],{"className":16935,"style":390},[388,389]," 上的 Morse 函数。当抬高阈值 ",[344,16938,16940,16953],{"className":16939,"translate":348},[347],[344,16941,16943],{"className":16942},[352],[354,16944,16945],{"xmlns":356},[358,16946,16947,16951],{},[361,16948,16949],{},[364,16950,5162],{},[368,16952,5162],{"encoding":370},[344,16954,16956],{"className":16955,"ariaHidden":375},[374],[344,16957,16959,16962],{"className":16958},[379],[344,16960],{"className":16961,"style":865},[383],[344,16963,5162],{"className":16964},[388,389]," 并考察下水平集",[344,16967,16969],{"className":16968,"translate":348},[1047],[344,16970,16972,17018],{"className":16971,"translate":348},[347],[344,16973,16975],{"className":16974},[352],[354,16976,16977],{"xmlns":356,"display":1056},[358,16978,16979,17015],{},[361,16980,16981,16987,16989,16991,16993,16995,16997,16999,17001,17003,17005,17007,17009,17011,17013],{},[822,16982,16983,16985],{},[364,16984,16851],{},[364,16986,5162],{},[450,16988,452],{},[450,16990,818],{},[450,16992,6863],{"stretchy":592},[364,16994,815],{},[450,16996,580],{},[364,16998,16851],{},[450,17000,452],{},[364,17002,448],{},[450,17004,593],{"stretchy":592},[364,17006,815],{},[450,17008,602],{"stretchy":592},[450,17010,15011],{},[364,17012,5162],{},[450,17014,6881],{"stretchy":592},[368,17016,17017],{"encoding":370},"M_a := \\{x\\in M : f(x)\\le a\\}",[344,17019,17021,17077,17098,17116,17143],{"className":17020,"ariaHidden":375},[374],[344,17022,17024,17027,17068,17071,17074],{"className":17023},[379],[344,17025],{"className":17026,"style":12324},[383],[344,17028,17030,17033],{"className":17029},[388],[344,17031,16851],{"className":17032,"style":390},[388,389],[344,17034,17036],{"className":17035},[513],[344,17037,17039,17060],{"className":17038},[517,900],[344,17040,17042,17057],{"className":17041},[521],[344,17043,17045],{"className":17044,"style":1761},[525],[344,17046,17048,17051],{"style":17047},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[344,17049],{"className":17050,"style":533},[532],[344,17052,17054],{"className":17053},[537,538,539,540],[344,17055,5162],{"className":17056},[388,389,540],[344,17058,924],{"className":17059},[923],[344,17061,17063],{"className":17062},[521],[344,17064,17066],{"className":17065,"style":931},[525],[344,17067],{},[344,17069],{"className":17070,"style":488},[487],[344,17072,1226],{"className":17073},[492],[344,17075],{"className":17076,"style":488},[487],[344,17078,17080,17083,17086,17089,17092,17095],{"className":17079},[379],[344,17081],{"className":17082,"style":884},[383],[344,17084,6863],{"className":17085},[668],[344,17087,815],{"className":17088},[388,389],[344,17090],{"className":17091,"style":488},[487],[344,17093,580],{"className":17094},[492],[344,17096],{"className":17097,"style":488},[487],[344,17099,17101,17104,17107,17110,17113],{"className":17100},[379],[344,17102],{"className":17103,"style":384},[383],[344,17105,16851],{"className":17106,"style":390},[388,389],[344,17108],{"className":17109,"style":488},[487],[344,17111,452],{"className":17112},[492],[344,17114],{"className":17115,"style":488},[487],[344,17117,17119,17122,17125,17128,17131,17134,17137,17140],{"className":17118},[379],[344,17120],{"className":17121,"style":884},[383],[344,17123,448],{"className":17124,"style":483},[388,389],[344,17126,593],{"className":17127},[668],[344,17129,815],{"className":17130},[388,389],[344,17132,602],{"className":17133},[701],[344,17135],{"className":17136,"style":488},[487],[344,17138,15011],{"className":17139},[492],[344,17141],{"className":17142,"style":488},[487],[344,17144,17146,17149,17152],{"className":17145},[379],[344,17147],{"className":17148,"style":884},[383],[344,17150,5162],{"className":17151},[388,389],[344,17153,6881],{"className":17154},[701],[15,17156,17157,17158,17228,17229,17257,17258,17286,17287,17315],{},"时，",[344,17159,17161,17179],{"className":17160,"translate":348},[347],[344,17162,17164],{"className":17163},[352],[354,17165,17166],{"xmlns":356},[358,17167,17168,17176],{},[361,17169,17170],{},[822,17171,17172,17174],{},[364,17173,16851],{},[364,17175,5162],{},[368,17177,17178],{"encoding":370},"M_a",[344,17180,17182],{"className":17181,"ariaHidden":375},[374],[344,17183,17185,17188],{"className":17184},[379],[344,17186],{"className":17187,"style":12324},[383],[344,17189,17191,17194],{"className":17190},[388],[344,17192,16851],{"className":17193,"style":390},[388,389],[344,17195,17197],{"className":17196},[513],[344,17198,17200,17220],{"className":17199},[517,900],[344,17201,17203,17217],{"className":17202},[521],[344,17204,17206],{"className":17205,"style":1761},[525],[344,17207,17208,17211],{"style":17047},[344,17209],{"className":17210,"style":533},[532],[344,17212,17214],{"className":17213},[537,538,539,540],[344,17215,5162],{"className":17216},[388,389,540],[344,17218,924],{"className":17219},[923],[344,17221,17223],{"className":17222},[521],[344,17224,17226],{"className":17225,"style":931},[525],[344,17227],{}," 的拓扑只在 ",[344,17230,17232,17245],{"className":17231,"translate":348},[347],[344,17233,17235],{"className":17234},[352],[354,17236,17237],{"xmlns":356},[358,17238,17239,17243],{},[361,17240,17241],{},[364,17242,5162],{},[368,17244,5162],{"encoding":370},[344,17246,17248],{"className":17247,"ariaHidden":375},[374],[344,17249,17251,17254],{"className":17250},[379],[344,17252],{"className":17253,"style":865},[383],[344,17255,5162],{"className":17256},[388,389]," 经过临界值时改变。若临界点的指数为 ",[344,17259,17261,17274],{"className":17260,"translate":348},[347],[344,17262,17264],{"className":17263},[352],[354,17265,17266],{"xmlns":356},[358,17267,17268,17272],{},[361,17269,17270],{},[364,17271,15014],{},[368,17273,15014],{"encoding":370},[344,17275,17277],{"className":17276,"ariaHidden":375},[374],[344,17278,17280,17283],{"className":17279},[379],[344,17281],{"className":17282,"style":3391},[383],[344,17284,15014],{"className":17285,"style":15054},[388,389],"，那么粗略地说，穿过该水平会连接一个 ",[344,17288,17290,17303],{"className":17289,"translate":348},[347],[344,17291,17293],{"className":17292},[352],[354,17294,17295],{"xmlns":356},[358,17296,17297,17301],{},[361,17298,17299],{},[364,17300,15014],{},[368,17302,15014],{"encoding":370},[344,17304,17306],{"className":17305,"ariaHidden":375},[374],[344,17307,17309,17312],{"className":17308},[379],[344,17310],{"className":17311,"style":3391},[383],[344,17313,15014],{"className":17314,"style":15054},[388,389]," 维手柄：局部极小值的指数为 0，因此它会为下水平集创造出新的连通分支。指数为 1 的鞍点可以把分支合并。更高指数的鞍点则创造出更高维的隧道与空洞。",[15,17317,17318,17319,8905],{},"一个推论是欧拉示性数的 ",[324,17320,17321],{},"Morse 关系",[344,17323,17325],{"className":17324,"translate":348},[1047],[344,17326,17328,17400],{"className":17327,"translate":348},[347],[344,17329,17331],{"className":17330},[352],[354,17332,17333],{"xmlns":356,"display":1056},[358,17334,17335,17397],{},[1082,17336,17337],{"width":3486},[1088,17338,17339,17341,17390,17392],{},[1091,17340],{"width":3491},[1091,17342,17343],{},[361,17344,17345,17359,17361,17363,17365,17371,17377,17379,17382,17384,17386,17388],{},[5143,17346,17347,17349,17357],{},[450,17348,5147],{},[361,17350,17351,17353,17355],{},[364,17352,15014],{},[450,17354,818],{},[587,17356,1096],{},[364,17358,366],{},[450,17360,593],{"stretchy":592},[450,17362,3372],{},[587,17364,828],{},[454,17366,17367,17369],{},[450,17368,602],{"stretchy":592},[364,17370,15014],{},[822,17372,17373,17375],{},[364,17374,585],{},[364,17376,15014],{},[450,17378,818],{},[364,17380,17381],{},"χ",[450,17383,593],{"stretchy":592},[364,17385,16851],{},[450,17387,602],{"stretchy":592},[450,17389,831],{"separator":375},[1091,17391],{"width":3491},[1091,17393,17394],{},[3609,17395,17396],{},"(4)",[368,17398,17399],{"encoding":370},"\\sum_{k=0}^{N} (-1)^k C_k = \\chi(M), \\tag{4}",[344,17401,17403,17567,17588],{"className":17402,"ariaHidden":375},[374],[344,17404,17406,17410,17479,17482,17485,17488,17518,17558,17561,17564],{"className":17405},[379],[344,17407],{"className":17408,"style":17409},[383],"height:3.1304em;vertical-align:-1.3021em;",[344,17411,17413],{"className":17412},[5207,5208],[344,17414,17416,17470],{"className":17415},[517,900],[344,17417,17419,17467],{"className":17418},[521],[344,17420,17422,17443,17453],{"className":17421,"style":5218},[525],[344,17423,17425,17428],{"style":17424},"top:-1.8479em;margin-left:0em;",[344,17426],{"className":17427,"style":5225},[532],[344,17429,17431],{"className":17430},[537,538,539,540],[344,17432,17434,17437,17440],{"className":17433},[388,540],[344,17435,15014],{"className":17436,"style":15054},[388,389,540],[344,17438,818],{"className":17439},[492,540],[344,17441,1096],{"className":17442},[388,540],[344,17444,17445,17448],{"style":5243},[344,17446],{"className":17447,"style":5225},[532],[344,17449,17450],{},[344,17451,5147],{"className":17452},[5207,5252,5253],[344,17454,17455,17458],{"style":5256},[344,17456],{"className":17457,"style":5225},[532],[344,17459,17461],{"className":17460},[537,538,539,540],[344,17462,17464],{"className":17463},[388,540],[344,17465,366],{"className":17466,"style":390},[388,389,540],[344,17468,924],{"className":17469},[923],[344,17471,17473],{"className":17472},[521],[344,17474,17477],{"className":17475,"style":17476},[525],"height:1.3021em;",[344,17478],{},[344,17480,593],{"className":17481},[668],[344,17483,3372],{"className":17484},[388],[344,17486,828],{"className":17487},[388],[344,17489,17491,17494],{"className":17490},[701],[344,17492,602],{"className":17493},[701],[344,17495,17497],{"className":17496},[513],[344,17498,17500],{"className":17499},[517],[344,17501,17503],{"className":17502},[521],[344,17504,17507],{"className":17505,"style":17506},[525],"height:0.8991em;",[344,17508,17509,17512],{"style":1682},[344,17510],{"className":17511,"style":533},[532],[344,17513,17515],{"className":17514},[537,538,539,540],[344,17516,15014],{"className":17517,"style":15054},[388,389,540],[344,17519,17521,17524],{"className":17520},[388],[344,17522,585],{"className":17523,"style":640},[388,389],[344,17525,17527],{"className":17526},[513],[344,17528,17530,17550],{"className":17529},[517,900],[344,17531,17533,17547],{"className":17532},[521],[344,17534,17536],{"className":17535,"style":2301},[525],[344,17537,17538,17541],{"style":15195},[344,17539],{"className":17540,"style":533},[532],[344,17542,17544],{"className":17543},[537,538,539,540],[344,17545,15014],{"className":17546,"style":15054},[388,389,540],[344,17548,924],{"className":17549},[923],[344,17551,17553],{"className":17552},[521],[344,17554,17556],{"className":17555,"style":931},[525],[344,17557],{},[344,17559],{"className":17560,"style":488},[487],[344,17562,818],{"className":17563},[492],[344,17565],{"className":17566,"style":488},[487],[344,17568,17570,17573,17576,17579,17582,17585],{"className":17569},[379],[344,17571],{"className":17572,"style":884},[383],[344,17574,17381],{"className":17575},[388,389],[344,17577,593],{"className":17578},[668],[344,17580,16851],{"className":17581,"style":390},[388,389],[344,17583,602],{"className":17584},[701],[344,17586,831],{"className":17587},[937],[344,17589,17591,17594],{"className":17590},[4116],[344,17592],{"className":17593,"style":17409},[383],[344,17595,17597,17600,17606],{"className":17596},[388,4123],[344,17598,593],{"className":17599},[388],[344,17601,17603],{"className":17602},[388],[344,17604,13106],{"className":17605},[388],[344,17607,602],{"className":17608},[388],[15,17610,17611,17612,17682,17683,17711,17712,8773,17756,17784,17785,17828],{},"其中 ",[344,17613,17615,17633],{"className":17614,"translate":348},[347],[344,17616,17618],{"className":17617},[352],[354,17619,17620],{"xmlns":356},[358,17621,17622,17630],{},[361,17623,17624],{},[822,17625,17626,17628],{},[364,17627,585],{},[364,17629,15014],{},[368,17631,17632],{"encoding":370},"C_k",[344,17634,17636],{"className":17635,"ariaHidden":375},[374],[344,17637,17639,17642],{"className":17638},[379],[344,17640],{"className":17641,"style":12324},[383],[344,17643,17645,17648],{"className":17644},[388],[344,17646,585],{"className":17647,"style":640},[388,389],[344,17649,17651],{"className":17650},[513],[344,17652,17654,17674],{"className":17653},[517,900],[344,17655,17657,17671],{"className":17656},[521],[344,17658,17660],{"className":17659,"style":2301},[525],[344,17661,17662,17665],{"style":15195},[344,17663],{"className":17664,"style":533},[532],[344,17666,17668],{"className":17667},[537,538,539,540],[344,17669,15014],{"className":17670,"style":15054},[388,389,540],[344,17672,924],{"className":17673},[923],[344,17675,17677],{"className":17676},[521],[344,17678,17680],{"className":17679,"style":931},[525],[344,17681],{}," 是指数为 ",[344,17684,17686,17699],{"className":17685,"translate":348},[347],[344,17687,17689],{"className":17688},[352],[354,17690,17691],{"xmlns":356},[358,17692,17693,17697],{},[361,17694,17695],{},[364,17696,15014],{},[368,17698,15014],{"encoding":370},[344,17700,17702],{"className":17701,"ariaHidden":375},[374],[344,17703,17705,17708],{"className":17704},[379],[344,17706],{"className":17707,"style":3391},[383],[344,17709,15014],{"className":17710,"style":15054},[388,389]," 的临界点个数，",[344,17713,17715,17735],{"className":17714,"translate":348},[347],[344,17716,17718],{"className":17717},[352],[354,17719,17720],{"xmlns":356},[358,17721,17722,17732],{},[361,17723,17724,17726,17728,17730],{},[364,17725,17381],{},[450,17727,593],{"stretchy":592},[364,17729,16851],{},[450,17731,602],{"stretchy":592},[368,17733,17734],{"encoding":370},"\\chi(M)",[344,17736,17738],{"className":17737,"ariaHidden":375},[374],[344,17739,17741,17744,17747,17750,17753],{"className":17740},[379],[344,17742],{"className":17743,"style":884},[383],[344,17745,17381],{"className":17746},[388,389],[344,17748,593],{"className":17749},[668],[344,17751,16851],{"className":17752,"style":390},[388,389],[344,17754,602],{"className":17755},[701],[344,17757,17759,17772],{"className":17758,"translate":348},[347],[344,17760,17762],{"className":17761},[352],[354,17763,17764],{"xmlns":356},[358,17765,17766,17770],{},[361,17767,17768],{},[364,17769,16851],{},[368,17771,16851],{"encoding":370},[344,17773,17775],{"className":17774,"ariaHidden":375},[374],[344,17776,17778,17781],{"className":17777},[379],[344,17779],{"className":17780,"style":384},[383],[344,17782,16851],{"className":17783,"style":390},[388,389]," 的欧拉示性数。由于在复杂的随机景观中，",[344,17786,17788,17807],{"className":17787,"translate":348},[347],[344,17789,17791],{"className":17790},[352],[354,17792,17793],{"xmlns":356},[358,17794,17795,17805],{},[361,17796,17797,17799,17801,17803],{},[364,17798,17381],{},[450,17800,593],{"stretchy":592},[364,17802,16851],{},[450,17804,602],{"stretchy":592},[368,17806,17734],{"encoding":370},[344,17808,17810],{"className":17809,"ariaHidden":375},[374],[344,17811,17813,17816,17819,17822,17825],{"className":17812},[379],[344,17814],{"className":17815,"style":884},[383],[344,17817,17381],{"className":17818},[388,389],[344,17820,593],{"className":17821},[668],[344,17823,16851],{"className":17824,"style":390},[388,389],[344,17826,602],{"className":17827},[701]," 通常相对于临界点总数很小，许多临界点必须在这个交错求和中相互抵消——相邻指数的鞍点天然适合这种抵消。",[15,17830,17831],{},"这一拓扑图景与概率图景一致：随机高维景观常常含有大量散布在各个指数上的鞍点。但重要的是不要夸大拓扑论证：球面上的函数可以只有一个极小值和一个极大值。Morse 理论约束着可能的临界点计数。随机矩阵理论与 Kac–Rice 则解释了为什么在随机高维模型中，计数的绝大部分通常是鞍点式的。",[11484,17833,17835],{"id":17834},"坐标不变性morse-引理","坐标不变性：Morse 引理",[15,17837,17838,17839,17842],{},"在非退化临界点附近，经过光滑坐标变换后，每个光滑函数都看起来像它的 Hessian 标准形——这正是 ",[324,17840,17841],{},"Morse 引理"," 的内容。",[764,17844,17845,18184],{},[15,17846,17847,5826,17850,17922,17923,17682,17975,18003,18004,18032,18033,18183],{},[324,17848,17849],{},"定理 B.1（Morse 引理）",[344,17851,17853,17874],{"className":17852,"translate":348},[347],[344,17854,17856],{"className":17855},[352],[354,17857,17858],{"xmlns":356},[358,17859,17860,17872],{},[361,17861,17862,17864,17866,17868,17870],{},[364,17863,448],{},[450,17865,452],{},[364,17867,16851],{},[450,17869,464],{},[364,17871,459],{"mathvariant":458},[368,17873,16858],{"encoding":370},[344,17875,17877,17895,17913],{"className":17876,"ariaHidden":375},[374],[344,17878,17880,17883,17886,17889,17892],{"className":17879},[379],[344,17881],{"className":17882,"style":479},[383],[344,17884,448],{"className":17885,"style":483},[388,389],[344,17887],{"className":17888,"style":488},[487],[344,17890,452],{"className":17891},[492],[344,17893],{"className":17894,"style":488},[487],[344,17896,17898,17901,17904,17907,17910],{"className":17897},[379],[344,17899],{"className":17900,"style":384},[383],[344,17902,16851],{"className":17903,"style":390},[388,389],[344,17905],{"className":17906,"style":488},[487],[344,17908,464],{"className":17909},[492],[344,17911],{"className":17912,"style":488},[487],[344,17914,17916,17919],{"className":17915},[379],[344,17917],{"className":17918,"style":559},[383],[344,17920,459],{"className":17921},[388,509]," 光滑，",[344,17924,17926,17944],{"className":17925,"translate":348},[347],[344,17927,17929],{"className":17928},[352],[354,17930,17931],{"xmlns":356},[358,17932,17933,17941],{},[361,17934,17935,17937,17939],{},[364,17936,15],{},[450,17938,580],{},[364,17940,16851],{},[368,17942,17943],{"encoding":370},"p\\in M",[344,17945,17947,17966],{"className":17946,"ariaHidden":375},[374],[344,17948,17950,17954,17957,17960,17963],{"className":17949},[379],[344,17951],{"className":17952,"style":17953},[383],"height:0.7335em;vertical-align:-0.1944em;",[344,17955,15],{"className":17956},[388,389],[344,17958],{"className":17959,"style":488},[487],[344,17961,580],{"className":17962},[492],[344,17964],{"className":17965,"style":488},[487],[344,17967,17969,17972],{"className":17968},[379],[344,17970],{"className":17971,"style":384},[383],[344,17973,16851],{"className":17974,"style":390},[388,389],[344,17976,17978,17991],{"className":17977,"translate":348},[347],[344,17979,17981],{"className":17980},[352],[354,17982,17983],{"xmlns":356},[358,17984,17985,17989],{},[361,17986,17987],{},[364,17988,15014],{},[368,17990,15014],{"encoding":370},[344,17992,17994],{"className":17993,"ariaHidden":375},[374],[344,17995,17997,18000],{"className":17996},[379],[344,17998],{"className":17999,"style":3391},[383],[344,18001,15014],{"className":18002,"style":15054},[388,389]," 的非退化临界点。则以 ",[344,18005,18007,18020],{"className":18006,"translate":348},[347],[344,18008,18010],{"className":18009},[352],[354,18011,18012],{"xmlns":356},[358,18013,18014,18018],{},[361,18015,18016],{},[364,18017,15],{},[368,18019,15],{"encoding":370},[344,18021,18023],{"className":18022,"ariaHidden":375},[374],[344,18024,18026,18029],{"className":18025},[379],[344,18027],{"className":18028,"style":4879},[383],[344,18030,15],{"className":18031},[388,389]," 为中心存在局部坐标 ",[344,18034,18036,18070],{"className":18035,"translate":348},[347],[344,18037,18039],{"className":18038},[352],[354,18040,18041],{"xmlns":356},[358,18042,18043,18067],{},[361,18044,18045,18047,18053,18055,18057,18059,18065],{},[450,18046,593],{"stretchy":592},[822,18048,18049,18051],{},[364,18050,9979],{},[587,18052,828],{},[450,18054,831],{"separator":375},[450,18056,834],{},[450,18058,831],{"separator":375},[822,18060,18061,18063],{},[364,18062,9979],{},[364,18064,366],{},[450,18066,602],{"stretchy":592},[368,18068,18069],{"encoding":370},"(u_1,\\dots,u_N)",[344,18071,18073],{"className":18072,"ariaHidden":375},[374],[344,18074,18076,18079,18082,18122,18125,18128,18131,18134,18137,18140,18180],{"className":18075},[379],[344,18077],{"className":18078,"style":884},[383],[344,18080,593],{"className":18081},[668],[344,18083,18085,18088],{"className":18084},[388],[344,18086,9979],{"className":18087},[388,389],[344,18089,18091],{"className":18090},[513],[344,18092,18094,18114],{"className":18093},[517,900],[344,18095,18097,18111],{"className":18096},[521],[344,18098,18100],{"className":18099,"style":907},[525],[344,18101,18102,18105],{"style":910},[344,18103],{"className":18104,"style":533},[532],[344,18106,18108],{"className":18107},[537,538,539,540],[344,18109,828],{"className":18110},[388,540],[344,18112,924],{"className":18113},[923],[344,18115,18117],{"className":18116},[521],[344,18118,18120],{"className":18119,"style":931},[525],[344,18121],{},[344,18123,831],{"className":18124},[937],[344,18126],{"className":18127,"style":941},[487],[344,18129,834],{"className":18130},[945],[344,18132],{"className":18133,"style":941},[487],[344,18135,831],{"className":18136},[937],[344,18138],{"className":18139,"style":941},[487],[344,18141,18143,18146],{"className":18142},[388],[344,18144,9979],{"className":18145},[388,389],[344,18147,18149],{"className":18148},[513],[344,18150,18152,18172],{"className":18151},[517,900],[344,18153,18155,18169],{"className":18154},[521],[344,18156,18158],{"className":18157,"style":973},[525],[344,18159,18160,18163],{"style":910},[344,18161],{"className":18162,"style":533},[532],[344,18164,18166],{"className":18165},[537,538,539,540],[344,18167,366],{"className":18168,"style":390},[388,389,540],[344,18170,924],{"className":18171},[923],[344,18173,18175],{"className":18174},[521],[344,18176,18178],{"className":18177,"style":931},[525],[344,18179],{},[344,18181,602],{"className":18182},[701],"，使得",[344,18185,18187],{"className":18186,"translate":348},[1047],[344,18188,18190,18277],{"className":18189,"translate":348},[347],[344,18191,18193],{"className":18192},[352],[354,18194,18195],{"xmlns":356,"display":1056},[358,18196,18197,18274],{},[361,18198,18199,18201,18203,18205,18207,18209,18211,18213,18215,18217,18219,18227,18229,18232,18234,18242,18244,18258,18260,18262,18264,18272],{},[364,18200,448],{},[450,18202,593],{"stretchy":592},[364,18204,9979],{},[450,18206,602],{"stretchy":592},[450,18208,818],{},[364,18210,448],{},[450,18212,593],{"stretchy":592},[364,18214,15],{},[450,18216,602],{"stretchy":592},[450,18218,3372],{},[2205,18220,18221,18223,18225],{},[364,18222,9979],{},[587,18224,828],{},[587,18226,589],{},[450,18228,3372],{},[450,18230,18231],{},"⋯",[450,18233,3372],{},[2205,18235,18236,18238,18240],{},[364,18237,9979],{},[364,18239,15014],{},[587,18241,589],{},[450,18243,3508],{},[2205,18245,18246,18248,18256],{},[364,18247,9979],{},[361,18249,18250,18252,18254],{},[364,18251,15014],{},[450,18253,3508],{},[587,18255,828],{},[587,18257,589],{},[450,18259,3508],{},[450,18261,18231],{},[450,18263,3508],{},[2205,18265,18266,18268,18270],{},[364,18267,9979],{},[364,18269,366],{},[587,18271,589],{},[364,18273,1192],{"mathvariant":1063},[368,18275,18276],{"encoding":370},"f(u) = f(p) - u_1^2 - \\dots - u_k^2 + u_{k+1}^2 + \\dots + u_N^2.",[344,18278,18280,18307,18334,18401,18419,18485,18562,18580],{"className":18279,"ariaHidden":375},[374],[344,18281,18283,18286,18289,18292,18295,18298,18301,18304],{"className":18282},[379],[344,18284],{"className":18285,"style":884},[383],[344,18287,448],{"className":18288,"style":483},[388,389],[344,18290,593],{"className":18291},[668],[344,18293,9979],{"className":18294},[388,389],[344,18296,602],{"className":18297},[701],[344,18299],{"className":18300,"style":488},[487],[344,18302,818],{"className":18303},[492],[344,18305],{"className":18306,"style":488},[487],[344,18308,18310,18313,18316,18319,18322,18325,18328,18331],{"className":18309},[379],[344,18311],{"className":18312,"style":884},[383],[344,18314,448],{"className":18315,"style":483},[388,389],[344,18317,593],{"className":18318},[668],[344,18320,15],{"className":18321},[388,389],[344,18323,602],{"className":18324},[701],[344,18326],{"className":18327,"style":2138},[487],[344,18329,3372],{"className":18330},[2142],[344,18332],{"className":18333,"style":2138},[487],[344,18335,18337,18341,18392,18395,18398],{"className":18336},[379],[344,18338],{"className":18339,"style":18340},[383],"height:1.1111em;vertical-align:-0.247em;",[344,18342,18344,18347],{"className":18343},[388],[344,18345,9979],{"className":18346},[388,389],[344,18348,18350],{"className":18349},[513],[344,18351,18353,18384],{"className":18352},[517,900],[344,18354,18356,18381],{"className":18355},[521],[344,18357,18359,18370],{"className":18358,"style":2369},[525],[344,18360,18361,18364],{"style":5726},[344,18362],{"className":18363,"style":533},[532],[344,18365,18367],{"className":18366},[537,538,539,540],[344,18368,828],{"className":18369},[388,540],[344,18371,18372,18375],{"style":1682},[344,18373],{"className":18374,"style":533},[532],[344,18376,18378],{"className":18377},[537,538,539,540],[344,18379,589],{"className":18380},[388,540],[344,18382,924],{"className":18383},[923],[344,18385,18387],{"className":18386},[521],[344,18388,18390],{"className":18389,"style":5756},[525],[344,18391],{},[344,18393],{"className":18394,"style":2138},[487],[344,18396,3372],{"className":18397},[2142],[344,18399],{"className":18400,"style":2138},[487],[344,18402,18404,18407,18410,18413,18416],{"className":18403},[379],[344,18405],{"className":18406,"style":3410},[383],[344,18408,18231],{"className":18409},[945],[344,18411],{"className":18412,"style":2138},[487],[344,18414,3372],{"className":18415},[2142],[344,18417],{"className":18418,"style":2138},[487],[344,18420,18422,18425,18476,18479,18482],{"className":18421},[379],[344,18423],{"className":18424,"style":18340},[383],[344,18426,18428,18431],{"className":18427},[388],[344,18429,9979],{"className":18430},[388,389],[344,18432,18434],{"className":18433},[513],[344,18435,18437,18468],{"className":18436},[517,900],[344,18438,18440,18465],{"className":18439},[521],[344,18441,18443,18454],{"className":18442,"style":2369},[525],[344,18444,18445,18448],{"style":5726},[344,18446],{"className":18447,"style":533},[532],[344,18449,18451],{"className":18450},[537,538,539,540],[344,18452,15014],{"className":18453,"style":15054},[388,389,540],[344,18455,18456,18459],{"style":1682},[344,18457],{"className":18458,"style":533},[532],[344,18460,18462],{"className":18461},[537,538,539,540],[344,18463,589],{"className":18464},[388,540],[344,18466,924],{"className":18467},[923],[344,18469,18471],{"className":18470},[521],[344,18472,18474],{"className":18473,"style":5756},[525],[344,18475],{},[344,18477],{"className":18478,"style":2138},[487],[344,18480,3508],{"className":18481},[2142],[344,18483],{"className":18484,"style":2138},[487],[344,18486,18488,18492,18553,18556,18559],{"className":18487},[379],[344,18489],{"className":18490,"style":18491},[383],"height:1.1694em;vertical-align:-0.3053em;",[344,18493,18495,18498],{"className":18494},[388],[344,18496,9979],{"className":18497},[388,389],[344,18499,18501],{"className":18500},[513],[344,18502,18504,18544],{"className":18503},[517,900],[344,18505,18507,18541],{"className":18506},[521],[344,18508,18510,18530],{"className":18509,"style":2369},[525],[344,18511,18512,18515],{"style":5726},[344,18513],{"className":18514,"style":533},[532],[344,18516,18518],{"className":18517},[537,538,539,540],[344,18519,18521,18524,18527],{"className":18520},[388,540],[344,18522,15014],{"className":18523,"style":15054},[388,389,540],[344,18525,3508],{"className":18526},[2142,540],[344,18528,828],{"className":18529},[388,540],[344,18531,18532,18535],{"style":1682},[344,18533],{"className":18534,"style":533},[532],[344,18536,18538],{"className":18537},[537,538,539,540],[344,18539,589],{"className":18540},[388,540],[344,18542,924],{"className":18543},[923],[344,18545,18547],{"className":18546},[521],[344,18548,18551],{"className":18549,"style":18550},[525],"height:0.3053em;",[344,18552],{},[344,18554],{"className":18555,"style":2138},[487],[344,18557,3508],{"className":18558},[2142],[344,18560],{"className":18561,"style":2138},[487],[344,18563,18565,18568,18571,18574,18577],{"className":18564},[379],[344,18566],{"className":18567,"style":3410},[383],[344,18569,18231],{"className":18570},[945],[344,18572],{"className":18573,"style":2138},[487],[344,18575,3508],{"className":18576},[2142],[344,18578],{"className":18579,"style":2138},[487],[344,18581,18583,18586,18637],{"className":18582},[379],[344,18584],{"className":18585,"style":18340},[383],[344,18587,18589,18592],{"className":18588},[388],[344,18590,9979],{"className":18591},[388,389],[344,18593,18595],{"className":18594},[513],[344,18596,18598,18629],{"className":18597},[517,900],[344,18599,18601,18626],{"className":18600},[521],[344,18602,18604,18615],{"className":18603,"style":2369},[525],[344,18605,18606,18609],{"style":5726},[344,18607],{"className":18608,"style":533},[532],[344,18610,18612],{"className":18611},[537,538,539,540],[344,18613,366],{"className":18614,"style":390},[388,389,540],[344,18616,18617,18620],{"style":1682},[344,18618],{"className":18619,"style":533},[532],[344,18621,18623],{"className":18622},[537,538,539,540],[344,18624,589],{"className":18625},[388,540],[344,18627,924],{"className":18628},[923],[344,18630,18632],{"className":18631},[521],[344,18633,18635],{"className":18634,"style":5756},[525],[344,18636],{},[344,18638,1192],{"className":18639},[388],[15,18641,18642],{},"因此，在光滑局部坐标变换意义下，非退化临界点完全由其指数分类。非退化局部极小值的标准形为",[344,18644,18646],{"className":18645,"translate":348},[1047],[344,18647,18649,18695],{"className":18648,"translate":348},[347],[344,18650,18652],{"className":18651},[352],[354,18653,18654],{"xmlns":356,"display":1056},[358,18655,18656,18692],{},[361,18657,18658,18660,18662,18664,18666,18668,18676,18678,18680,18682,18690],{},[364,18659,448],{},[450,18661,593],{"stretchy":592},[364,18663,15],{},[450,18665,602],{"stretchy":592},[450,18667,3508],{},[2205,18669,18670,18672,18674],{},[364,18671,9979],{},[587,18673,828],{},[587,18675,589],{},[450,18677,3508],{},[450,18679,18231],{},[450,18681,3508],{},[2205,18683,18684,18686,18688],{},[364,18685,9979],{},[364,18687,366],{},[587,18689,589],{},[450,18691,831],{"separator":375},[368,18693,18694],{"encoding":370},"f(p) + u_1^2 + \\dots + u_N^2,",[344,18696,18698,18725,18791,18809],{"className":18697,"ariaHidden":375},[374],[344,18699,18701,18704,18707,18710,18713,18716,18719,18722],{"className":18700},[379],[344,18702],{"className":18703,"style":884},[383],[344,18705,448],{"className":18706,"style":483},[388,389],[344,18708,593],{"className":18709},[668],[344,18711,15],{"className":18712},[388,389],[344,18714,602],{"className":18715},[701],[344,18717],{"className":18718,"style":2138},[487],[344,18720,3508],{"className":18721},[2142],[344,18723],{"className":18724,"style":2138},[487],[344,18726,18728,18731,18782,18785,18788],{"className":18727},[379],[344,18729],{"className":18730,"style":18340},[383],[344,18732,18734,18737],{"className":18733},[388],[344,18735,9979],{"className":18736},[388,389],[344,18738,18740],{"className":18739},[513],[344,18741,18743,18774],{"className":18742},[517,900],[344,18744,18746,18771],{"className":18745},[521],[344,18747,18749,18760],{"className":18748,"style":2369},[525],[344,18750,18751,18754],{"style":5726},[344,18752],{"className":18753,"style":533},[532],[344,18755,18757],{"className":18756},[537,538,539,540],[344,18758,828],{"className":18759},[388,540],[344,18761,18762,18765],{"style":1682},[344,18763],{"className":18764,"style":533},[532],[344,18766,18768],{"className":18767},[537,538,539,540],[344,18769,589],{"className":18770},[388,540],[344,18772,924],{"className":18773},[923],[344,18775,18777],{"className":18776},[521],[344,18778,18780],{"className":18779,"style":5756},[525],[344,18781],{},[344,18783],{"className":18784,"style":2138},[487],[344,18786,3508],{"className":18787},[2142],[344,18789],{"className":18790,"style":2138},[487],[344,18792,18794,18797,18800,18803,18806],{"className":18793},[379],[344,18795],{"className":18796,"style":3410},[383],[344,18798,18231],{"className":18799},[945],[344,18801],{"className":18802,"style":2138},[487],[344,18804,3508],{"className":18805},[2142],[344,18807],{"className":18808,"style":2138},[487],[344,18810,18812,18815,18866],{"className":18811},[379],[344,18813],{"className":18814,"style":18340},[383],[344,18816,18818,18821],{"className":18817},[388],[344,18819,9979],{"className":18820},[388,389],[344,18822,18824],{"className":18823},[513],[344,18825,18827,18858],{"className":18826},[517,900],[344,18828,18830,18855],{"className":18829},[521],[344,18831,18833,18844],{"className":18832,"style":2369},[525],[344,18834,18835,18838],{"style":5726},[344,18836],{"className":18837,"style":533},[532],[344,18839,18841],{"className":18840},[537,538,539,540],[344,18842,366],{"className":18843,"style":390},[388,389,540],[344,18845,18846,18849],{"style":1682},[344,18847],{"className":18848,"style":533},[532],[344,18850,18852],{"className":18851},[537,538,539,540],[344,18853,589],{"className":18854},[388,540],[344,18856,924],{"className":18857},[923],[344,18859,18861],{"className":18860},[521],[344,18862,18864],{"className":18863,"style":5756},[525],[344,18865],{},[344,18867,831],{"className":18868},[937],[15,18870,18871],{},"非退化局部极大值的标准形为",[344,18873,18875],{"className":18874,"translate":348},[1047],[344,18876,18878,18924],{"className":18877,"translate":348},[347],[344,18879,18881],{"className":18880},[352],[354,18882,18883],{"xmlns":356,"display":1056},[358,18884,18885,18921],{},[361,18886,18887,18889,18891,18893,18895,18897,18905,18907,18909,18911,18919],{},[364,18888,448],{},[450,18890,593],{"stretchy":592},[364,18892,15],{},[450,18894,602],{"stretchy":592},[450,18896,3372],{},[2205,18898,18899,18901,18903],{},[364,18900,9979],{},[587,18902,828],{},[587,18904,589],{},[450,18906,3372],{},[450,18908,18231],{},[450,18910,3372],{},[2205,18912,18913,18915,18917],{},[364,18914,9979],{},[364,18916,366],{},[587,18918,589],{},[450,18920,831],{"separator":375},[368,18922,18923],{"encoding":370},"f(p) - u_1^2 - \\dots - u_N^2,",[344,18925,18927,18954,19020,19038],{"className":18926,"ariaHidden":375},[374],[344,18928,18930,18933,18936,18939,18942,18945,18948,18951],{"className":18929},[379],[344,18931],{"className":18932,"style":884},[383],[344,18934,448],{"className":18935,"style":483},[388,389],[344,18937,593],{"className":18938},[668],[344,18940,15],{"className":18941},[388,389],[344,18943,602],{"className":18944},[701],[344,18946],{"className":18947,"style":2138},[487],[344,18949,3372],{"className":18950},[2142],[344,18952],{"className":18953,"style":2138},[487],[344,18955,18957,18960,19011,19014,19017],{"className":18956},[379],[344,18958],{"className":18959,"style":18340},[383],[344,18961,18963,18966],{"className":18962},[388],[344,18964,9979],{"className":18965},[388,389],[344,18967,18969],{"className":18968},[513],[344,18970,18972,19003],{"className":18971},[517,900],[344,18973,18975,19000],{"className":18974},[521],[344,18976,18978,18989],{"className":18977,"style":2369},[525],[344,18979,18980,18983],{"style":5726},[344,18981],{"className":18982,"style":533},[532],[344,18984,18986],{"className":18985},[537,538,539,540],[344,18987,828],{"className":18988},[388,540],[344,18990,18991,18994],{"style":1682},[344,18992],{"className":18993,"style":533},[532],[344,18995,18997],{"className":18996},[537,538,539,540],[344,18998,589],{"className":18999},[388,540],[344,19001,924],{"className":19002},[923],[344,19004,19006],{"className":19005},[521],[344,19007,19009],{"className":19008,"style":5756},[525],[344,19010],{},[344,19012],{"className":19013,"style":2138},[487],[344,19015,3372],{"className":19016},[2142],[344,19018],{"className":19019,"style":2138},[487],[344,19021,19023,19026,19029,19032,19035],{"className":19022},[379],[344,19024],{"className":19025,"style":3410},[383],[344,19027,18231],{"className":19028},[945],[344,19030],{"className":19031,"style":2138},[487],[344,19033,3372],{"className":19034},[2142],[344,19036],{"className":19037,"style":2138},[487],[344,19039,19041,19044,19095],{"className":19040},[379],[344,19042],{"className":19043,"style":18340},[383],[344,19045,19047,19050],{"className":19046},[388],[344,19048,9979],{"className":19049},[388,389],[344,19051,19053],{"className":19052},[513],[344,19054,19056,19087],{"className":19055},[517,900],[344,19057,19059,19084],{"className":19058},[521],[344,19060,19062,19073],{"className":19061,"style":2369},[525],[344,19063,19064,19067],{"style":5726},[344,19065],{"className":19066,"style":533},[532],[344,19068,19070],{"className":19069},[537,538,539,540],[344,19071,366],{"className":19072,"style":390},[388,389,540],[344,19074,19075,19078],{"style":1682},[344,19076],{"className":19077,"style":533},[532],[344,19079,19081],{"className":19080},[537,538,539,540],[344,19082,589],{"className":19083},[388,540],[344,19085,924],{"className":19086},[923],[344,19088,19090],{"className":19089},[521],[344,19091,19093],{"className":19092,"style":5756},[525],[344,19094],{},[344,19096,831],{"className":19097},[937],[15,19099,19100,19101,19129,19130,19158,19159,19210],{},"而指数为 ",[344,19102,19104,19117],{"className":19103,"translate":348},[347],[344,19105,19107],{"className":19106},[352],[354,19108,19109],{"xmlns":356},[358,19110,19111,19115],{},[361,19112,19113],{},[364,19114,15014],{},[368,19116,15014],{"encoding":370},[344,19118,19120],{"className":19119,"ariaHidden":375},[374],[344,19121,19123,19126],{"className":19122},[379],[344,19124],{"className":19125,"style":3391},[383],[344,19127,15014],{"className":19128,"style":15054},[388,389]," 的鞍点有 ",[344,19131,19133,19146],{"className":19132,"translate":348},[347],[344,19134,19136],{"className":19135},[352],[354,19137,19138],{"xmlns":356},[358,19139,19140,19144],{},[361,19141,19142],{},[364,19143,15014],{},[368,19145,15014],{"encoding":370},[344,19147,19149],{"className":19148,"ariaHidden":375},[374],[344,19150,19152,19155],{"className":19151},[379],[344,19153],{"className":19154,"style":3391},[383],[344,19156,15014],{"className":19157,"style":15054},[388,389]," 个向下的方向和 ",[344,19160,19162,19180],{"className":19161,"translate":348},[347],[344,19163,19165],{"className":19164},[352],[354,19166,19167],{"xmlns":356},[358,19168,19169,19177],{},[361,19170,19171,19173,19175],{},[364,19172,366],{},[450,19174,3372],{},[364,19176,15014],{},[368,19178,19179],{"encoding":370},"N-k",[344,19181,19183,19201],{"className":19182,"ariaHidden":375},[374],[344,19184,19186,19189,19192,19195,19198],{"className":19185},[379],[344,19187],{"className":19188,"style":2131},[383],[344,19190,366],{"className":19191,"style":390},[388,389],[344,19193],{"className":19194,"style":2138},[487],[344,19196,3372],{"className":19197},[2142],[344,19199],{"className":19200,"style":2138},[487],[344,19202,19204,19207],{"className":19203},[379],[344,19205],{"className":19206,"style":3391},[383],[344,19208,15014],{"className":19209,"style":15054},[388,389]," 个向上的方向。这正是「指数」成为高维临界点自然语言的原因。",[11,19212,19213],{"id":19213},"局限性",[15,19215,19216],{},"任何听起来很「美」的结论都需要打上补丁，这个也不例外。",[314,19218,19219,19391,19545,19551],{},[317,19220,19221,19224,19225,19283,19284,19390],{},[324,19222,19223],{},"不是所有函数都适用。"," ",[344,19226,19228,19245],{"className":19227,"translate":348},[347],[344,19229,19231],{"className":19230},[352],[354,19232,19233],{"xmlns":356},[358,19234,19235,19243],{},[361,19236,19237],{},[454,19238,19239,19241],{},[364,19240,459],{"mathvariant":458},[364,19242,366],{},[368,19244,8023],{"encoding":370},[344,19246,19248],{"className":19247,"ariaHidden":375},[374],[344,19249,19251,19254],{"className":19250},[379],[344,19252],{"className":19253,"style":502},[383],[344,19255,19257,19260],{"className":19256},[388],[344,19258,459],{"className":19259},[388,509],[344,19261,19263],{"className":19262},[513],[344,19264,19266],{"className":19265},[517],[344,19267,19269],{"className":19268},[521],[344,19270,19272],{"className":19271,"style":502},[525],[344,19273,19274,19277],{"style":528},[344,19275],{"className":19276,"style":533},[532],[344,19278,19280],{"className":19279},[537,538,539,540],[344,19281,366],{"className":19282,"style":390},[388,389,540]," 上的强凸函数，例如 ",[344,19285,19287,19319],{"className":19286,"translate":348},[347],[344,19288,19290],{"className":19289},[352],[354,19291,19292],{"xmlns":356},[358,19293,19294,19316],{},[361,19295,19296,19298,19300,19302,19304,19306,19308,19310],{},[364,19297,448],{},[450,19299,593],{"stretchy":592},[364,19301,815],{},[450,19303,602],{"stretchy":592},[450,19305,818],{},[364,19307,3591],{"mathvariant":1063},[364,19309,815],{},[454,19311,19312,19314],{},[364,19313,3591],{"mathvariant":1063},[587,19315,589],{},[368,19317,19318],{"encoding":370},"f(x)=\\|x\\|^2",[344,19320,19322,19349],{"className":19321,"ariaHidden":375},[374],[344,19323,19325,19328,19331,19334,19337,19340,19343,19346],{"className":19324},[379],[344,19326],{"className":19327,"style":884},[383],[344,19329,448],{"className":19330,"style":483},[388,389],[344,19332,593],{"className":19333},[668],[344,19335,815],{"className":19336},[388,389],[344,19338,602],{"className":19339},[701],[344,19341],{"className":19342,"style":488},[487],[344,19344,818],{"className":19345},[492],[344,19347],{"className":19348,"style":488},[487],[344,19350,19352,19355,19358,19361],{"className":19351},[379],[344,19353],{"className":19354,"style":13016},[383],[344,19356,3591],{"className":19357},[388],[344,19359,815],{"className":19360},[388,389],[344,19362,19364,19367],{"className":19363},[388],[344,19365,3591],{"className":19366},[388],[344,19368,19370],{"className":19369},[513],[344,19371,19373],{"className":19372},[517],[344,19374,19376],{"className":19375},[521],[344,19377,19379],{"className":19378,"style":653},[525],[344,19380,19381,19384],{"style":528},[344,19382],{"className":19383,"style":533},[532],[344,19385,19387],{"className":19386},[537,538,539,540],[344,19388,589],{"className":19389},[388,540],"，只有一个临界点，而且它是严格的全局极小值，没有鞍点。因此，仅仅高维并不意味着极小值稀缺——起决定作用的是 Hessian 有没有系统性地偏向正定。",[317,19392,19393,19396,19397,800,19493,19544],{},[324,19394,19395],{},"正定是严格极小值的充分条件，而非必要条件。"," 退化的严格极小值是存在的：一维函数 ",[344,19398,19400,19428],{"className":19399,"translate":348},[347],[344,19401,19403],{"className":19402},[352],[354,19404,19405],{"xmlns":356},[358,19406,19407,19425],{},[361,19408,19409,19411,19413,19415,19417,19419],{},[364,19410,448],{},[450,19412,593],{"stretchy":592},[364,19414,815],{},[450,19416,602],{"stretchy":592},[450,19418,818],{},[454,19420,19421,19423],{},[364,19422,815],{},[587,19424,13106],{},[368,19426,19427],{"encoding":370},"f(x)=x^4",[344,19429,19431,19458],{"className":19430,"ariaHidden":375},[374],[344,19432,19434,19437,19440,19443,19446,19449,19452,19455],{"className":19433},[379],[344,19435],{"className":19436,"style":884},[383],[344,19438,448],{"className":19439,"style":483},[388,389],[344,19441,593],{"className":19442},[668],[344,19444,815],{"className":19445},[388,389],[344,19447,602],{"className":19448},[701],[344,19450],{"className":19451,"style":488},[487],[344,19453,818],{"className":19454},[492],[344,19456],{"className":19457,"style":488},[487],[344,19459,19461,19464],{"className":19460},[379],[344,19462],{"className":19463,"style":653},[383],[344,19465,19467,19470],{"className":19466},[388],[344,19468,815],{"className":19469},[388,389],[344,19471,19473],{"className":19472},[513],[344,19474,19476],{"className":19475},[517],[344,19477,19479],{"className":19478},[521],[344,19480,19482],{"className":19481,"style":653},[525],[344,19483,19484,19487],{"style":528},[344,19485],{"className":19486,"style":533},[532],[344,19488,19490],{"className":19489},[537,538,539,540],[344,19491,13106],{"className":19492},[388,540],[344,19494,19496,19514],{"className":19495,"translate":348},[347],[344,19497,19499],{"className":19498},[352],[354,19500,19501],{"xmlns":356},[358,19502,19503,19511],{},[361,19504,19505,19507,19509],{},[364,19506,815],{},[450,19508,818],{},[587,19510,1096],{},[368,19512,19513],{"encoding":370},"x=0",[344,19515,19517,19535],{"className":19516,"ariaHidden":375},[374],[344,19518,19520,19523,19526,19529,19532],{"className":19519},[379],[344,19521],{"className":19522,"style":865},[383],[344,19524,815],{"className":19525},[388,389],[344,19527],{"className":19528,"style":488},[487],[344,19530,818],{"className":19531},[492],[344,19533],{"className":19534,"style":488},[487],[344,19536,19538,19541],{"className":19537},[379],[344,19539],{"className":19540,"style":7133},[383],[344,19542,1096],{"className":19543},[388]," 处有严格局部极小值，但其 Hessian 在那里为零。因此，Hessian 符号论证针对的是非退化的、即 Morse 的临界点。",[317,19546,19547,19550],{},[324,19548,19549],{},"结论依赖于随机景观系综。"," 中心化、类 GOE 的 Hessian 导致鞍点主导。Hessian 强烈向正方向平移的模型可以有许多极小值。在高斯场模型中，以极低函数值为条件，可以把 Hessian 谱向上平移，使极小值在景观底部附近更可能出现。",[317,19552,19553,19556,19557,19585],{},[324,19554,19555],{},"罕见并不意味着不存在。"," 即使极小值在所有临界点中的占比趋于零，在某些模型中极小值的绝对数目仍可能随 ",[344,19558,19560,19573],{"className":19559,"translate":348},[347],[344,19561,19563],{"className":19562},[352],[354,19564,19565],{"xmlns":356},[358,19566,19567,19571],{},[361,19568,19569],{},[364,19570,366],{},[368,19572,366],{"encoding":370},[344,19574,19576],{"className":19575,"ariaHidden":375},[374],[344,19577,19579,19582],{"className":19578},[379],[344,19580],{"className":19581,"style":384},[383],[344,19583,366],{"className":19584,"style":390},[388,389]," 增长。变得罕见的只是「随机选出的临界点是极小值」这一概率。",[15,19587,19588,19589,19592],{},"有了这些限定，可以更准确地表述为：",[324,19590,19591],{},"在许多 Hessian 谱近似无偏的高维随机景观中，一个典型的非退化临界点压倒性地倾向于成为鞍点。"," Morse 临界点中的局部极小值要求所有 Hessian 特征值为正，而这一正定事件随维度增长呈指数级罕见。",[11,19594,19595],{"id":19595},"优化实践的启发",[15,19597,19598],{},"这有助于解释为什么高维优化常被描述为「鞍点逃逸问题」，而非单纯的「坏局部极小值问题」。这一陈述不应被过度解读：它并不证明梯度下降总是有效，也不是说所有实际目标都表现得像 GOE 随机矩阵——真实的损失函数具有架构、数据、对称性、退化与约束，远比理想化模型复杂。",[15,19600,19601],{},"尽管如此，几何上的教训是有用的：",[314,19603,19604,19607,19610],{},[317,19605,19606],{},"高维中，绝大多数临界点是鞍点，其负特征值方向提供了逃离的「出口」（负曲率方向）。",[317,19608,19609],{},"因此，优化卡住的主因不是「掉进局部极小值的陷阱」，而是在高原或鞍点附近「停滞」。",[317,19611,19612],{},"算法如 SGD（随机梯度下降）自带的噪声，恰好能高效地把优化过程推离这些鞍点。而二阶算法（如牛顿法）反而需要谨慎处理负曲率方向，否则容易被带偏。",[15,19614,19615],{},"最后提醒一点：上述结论适用于随机、无结构化的景观。在深度学习中，由于对称性（如神经元置换）和平坦极小值的存在，景观结构更加复杂，但「鞍点主导」这一高维几何直觉依然成立。",[11,19617,19618],{"id":19618},"结论",[15,19620,19621,19649,19650,19678],{},[344,19622,19624,19637],{"className":19623,"translate":348},[347],[344,19625,19627],{"className":19626},[352],[354,19628,19629],{"xmlns":356},[358,19630,19631,19635],{},[361,19632,19633],{},[364,19634,366],{},[368,19636,366],{"encoding":370},[344,19638,19640],{"className":19639,"ariaHidden":375},[374],[344,19641,19643,19646],{"className":19642},[379],[344,19644],{"className":19645,"style":384},[383],[344,19647,366],{"className":19648,"style":390},[388,389]," 维中的非退化局部极小值必须通过 ",[344,19651,19653,19666],{"className":19652,"translate":348},[347],[344,19654,19656],{"className":19655},[352],[354,19657,19658],{"xmlns":356},[358,19659,19660,19664],{},[361,19661,19662],{},[364,19663,366],{},[368,19665,366],{"encoding":370},[344,19667,19669],{"className":19668,"ariaHidden":375},[374],[344,19670,19672,19675],{"className":19671},[379],[344,19673],{"className":19674,"style":384},[383],[344,19676,366],{"className":19677,"style":390},[388,389]," 个同时成立的曲率检验——在临界点处，每个 Hessian 特征值都必须为正。鞍点只需要正负号的混合。随机矩阵理论表明，大型中心化对称矩阵天然具有这样的混合：它们的谱散布在零附近，有大量正、负特征值。「所有特征值都落在正侧」这一事件是一个大偏差。",[15,19680,19681,19682,19710,19711,19739],{},"这解释了高维的直觉。临界点本身是光滑景观中体积为零的特征，但在临界点之中，决定性的问题是 Hessian 的符号。在随机模型中，典型的符号其指数靠近中间，而不是极端：非退化局部极小值指数为 0，非退化局部极大值指数为 ",[344,19683,19685,19698],{"className":19684,"translate":348},[347],[344,19686,19688],{"className":19687},[352],[354,19689,19690],{"xmlns":356},[358,19691,19692,19696],{},[361,19693,19694],{},[364,19695,366],{},[368,19697,366],{"encoding":370},[344,19699,19701],{"className":19700,"ariaHidden":375},[374],[344,19702,19704,19707],{"className":19703},[379],[344,19705],{"className":19706,"style":384},[383],[344,19708,366],{"className":19709,"style":390},[388,389],"，鞍点占据两者之间的所有指数。随着 ",[344,19712,19714,19727],{"className":19713,"translate":348},[347],[344,19715,19717],{"className":19716},[352],[354,19718,19719],{"xmlns":356},[358,19720,19721,19725],{},[361,19722,19723],{},[364,19724,366],{},[368,19726,366],{"encoding":370},[344,19728,19730],{"className":19729,"ariaHidden":375},[374],[344,19731,19733,19736],{"className":19732},[379],[344,19734],{"className":19735,"style":384},[383],[344,19737,366],{"className":19738,"style":390},[388,389]," 增长，中间压倒了极端。",[15,19741,19742],{},"因此，最好把这一观点理解为一条稳健的概率原则，而非关于每个高维函数的定理：",[764,19744,19745],{},[15,19746,19747],{},"在高维随机景观中，指数为零的 Morse 临界点是例外，而鞍点才是默认。",[15,19749,19750],{},"这正是深度学习等现代优化能够成功的理论基石之一：绝大多数临界点是鞍点，其负曲率方向提供了逃离的出口。带噪声的一阶方法能高效地把优化过程推开这些鞍点。",[19752,19753,19754],"h4",{"id":19754},"参考",[15,19756,19757,19758,19764],{},"参考来源：",[5162,19759,19763],{"href":19760,"rel":19761},"https:\u002F\u002Fwww.grantstenger.com\u002Flocal-minima",[19762],"nofollow","《Why Local Minima Are Rare in High-Dimensional Landscapes》","，Grant Stenger.",{"title":278,"searchDepth":279,"depth":279,"links":19766},[19767,19768,19769,19770,19771,19777,19778,19781,19782,19783],{"id":336,"depth":279,"text":336},{"id":429,"depth":279,"text":430},{"id":7485,"depth":279,"text":7485},{"id":8916,"depth":279,"text":8916},{"id":11476,"depth":279,"text":11476,"children":19772},[19773,19775,19776],{"id":11486,"depth":19774,"text":11486},3,{"id":11871,"depth":19774,"text":11872},{"id":13443,"depth":19774,"text":13443},{"id":14872,"depth":279,"text":14873},{"id":16826,"depth":279,"text":16827,"children":19779},[19780],{"id":17834,"depth":19774,"text":17835},{"id":19213,"depth":279,"text":19213},{"id":19595,"depth":279,"text":19595},{"id":19618,"depth":279,"text":19618},"兰州",{},"\u002Fblog\u002F2026\u002F2026-08-22-are-local-minima-rare-in-high-dimensions",{"title":304,"description":309},"blog\u002F2026\u002F2026-08-22-are-local-minima-rare-in-high-dimensions","高维空间中的局部极小值极为罕见——因为一个真正的局部极小值要求所有方向都向上弯曲，也就是 Hessian 矩阵的所有特征值都必须为正。而高维空间里，随机的临界点几乎总有某个方向可以向下走，也就是鞍点。这正是为什么神经网络训练更像是「逃离鞍点」的游戏，而不是「跳出局部最优」的游戏。",[19791],"machine-learning","2026-08-22T14:32:17+08:00","T8eGeIi0CA1m4XaOFzLI17zaHqy_AQWibRj-YAZ432c",{"left":19795,"top":19795,"width":19796,"height":19796,"rotate":19795,"vFlip":289,"hFlip":289,"body":19797},0,24,"\u003Cg fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"round\" stroke-linejoin=\"round\" stroke-width=\"2\">\u003Cpath d=\"m4.748 7.645l-.331 3.644c-.05.54-.074.811-.03 1.07a2 2 0 0 0 .238.655c.131.228.325.422.711.808l5.176 5.176c.787.787 1.18 1.18 1.636 1.328c.402.131.835.131 1.237 0c.456-.148.853-.544 1.645-1.336l3.96-3.96c.792-.792 1.187-1.188 1.336-1.644a2 2 0 0 0-.001-1.236c-.148-.457-.543-.853-1.335-1.645l-5.163-5.163c-.39-.39-.584-.584-.813-.716a2 2 0 0 0-.656-.238c-.26-.045-.535-.02-1.084.03l-3.63.33c-.944.086-1.417.129-1.787.335a2 2 0 0 0-.775.775c-.205.368-.248.838-.333 1.773z\"\u002F>\u003Cpath d=\"M9.713 9.713a1 1 0 1 0-1.415-1.414a1 1 0 0 0 1.415 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