The curse of dimensionality is not a specific geometric structure, but a collection of phenomena induced by the intrinsic properties of high-dimensional spaces. In particular, effects such as concentration of measure and distance concentration lead to the degeneration of metric discriminability, making many distance-based methods ineffective.

This article introduces the Curse of Dimensionality effect in spaces of different curvature from mathematical and statistical perspectives.

Euclidean Space

To introduce the curse of dimensionality, we need to start with the sample distribution in space.

Let us start with $1$-dimension, a line segment in the numerical range $[-1,1]$, and assume that all samples follow a random and uniform distribution on this line segment. Also, the unit measure of the line segment is $1$ unit of length.

Now, we consider the inner region that contains 90% of the measure, i.e., the interval $[-0.9, 0.9]$, and treat the remaining 10% as a thin outer shell with finite thickness. It is evident that 90% of the samples are randomly distributed within $[-0.9, 0.9]$, while the remaining 10% are located in the shell regions, namely $[-1, -0.9)$ and $(0.9, 1]$.

Obviously, we can infer that the proportion of samples distributed on the thin shell in $1$-dimension is $P_\text{shell}^{d=1} = 10\%$.

Now, let us extend this similar situation to circles in a $2$-dimensional plane. As we all know, the area of a circle can be calculated with the formula $S = \pi r^2$. Also, we consider the inner region that contains 90% of the measure, i.e., $x^2 + y^2 \leq 0.9$, and the remaining finite shell $0.9 < x^2 + y^2 \leq 1$. Next, we need to calculate the proportion of samples distributed in these two regions.

Similarly, we can continue our exploration in $3$-dimensional space. The volume for a $3$-dimensional sphere is $V=\frac{4}{3} \pi r^3$.

Here, we seem to be able to summarize a general pattern: as the dimension increases, the proportion of samples distributed on the outer shell of the space increases continuously $10\% \rightarrow 27.1\%$, while the proportion of samples in the cavity decreases continuously $90\% \rightarrow 72.9\%$.

To rigorously verify this conclusion, we need general mathematical reasoning. For a $d$-dimensional space, the volume of a hypersphere with radius $r$ is:

Where:

• $\Gamma(\cdot)$ is the gamma function;
• When $d$ is a positive integer, $\Gamma\left(\frac{d}{2}+1\right) = \left(\frac{d}{2}\right)!$, if $d$ is even;
• If $d$ is odd, the gamma function gives an expression containing $\sqrt{\pi}$.

We have:

For high dimensions $d \rightarrow +\infty$, we have the limit:

That is to say, random and uniform samples tend to distribute on the outer shell as the dimension $d$ grows. This is named the Thin-Shell Effect.

Euclidean Distance

Consider a high-dimensional space $\mathbb{R}^d$ with random vectors:

The magnitude of the vector $x$ is:

Since $x_i^2 \sim \chi^2(1)$, we have:

Here, $\chi^2(k)$ is the Chi-Squared Distribution, defined as the sum of squares of $k$ mutually independent random variables following a standard normal distribution $\mathcal{N}(0,1)$. In this case, since the squared norm is the sum of $d$ such variables, the degrees of freedom is $k=d$.

According to the Law of Large Numbers:

That is, the magnitude of vectors is approximated by $||x|| \approx \sqrt{d}$.

We have the Euclidean distance:

We have known:

The inner product of vectors $x, y$ is:

Since:

• $x_i, y_i$ are independent
• $E[x_i y_i] = 0$
• $\text{Var}(x_i y_i) = 1$

Therefore:

The order of magnitude is $O(\sqrt{d})$.

Substituting back into the distance equation, we have:

Thus:

Looking at the relative fluctuation:

• Mean: $2d$
• Standard Deviation: $O(\sqrt{d})$

Therefore:

Conclusion: The relative fluctuation of distance tends to 0 $\rightarrow$ the distance between all points is almost the same.

Angular Distance

If we project vectors onto the unit hypersphere after $\ell_2$ normalization:

Then:

In high dimensions:

Therefore:

The angular distance between all vectors is almost equidistant.

In high-dimensional spaces, the norm of random vectors concentrates around a constant due to the law of large numbers. Moreover, the inner product between independent vectors grows only on the order of $O(\sqrt{d})$, which is negligible compared to the $O(d)$ magnitude of squared norms. As a result, pairwise distances concentrate around a constant, leading to the phenomenon that almost all points are approximately equidistant.

Positively Curved Riemannian Space

In Euclidean space, the curse of dimensionality arises from the polynomial growth of volume and the independence structure of coordinates. However, in a positively curved Riemannian manifold, the geometry itself fundamentally reshapes both volume distribution and distance behavior.

A canonical example is the $d$-dimensional unit sphere:

which is a space with constant positive sectional curvature.

Measure Concentration on the Sphere

Unlike Euclidean space, where volume spreads radially, on the sphere all points lie on a fixed-radius manifold. Thus, “radial shells” are replaced by geodesic bands.

Let us fix a point $x_0 \in \mathbb{S}^{d-1}$ and define a geodesic ball:

The measure of this region depends on $\sin^{d-2}(\theta)$:

Now observe the key phenomenon:

• When $d$ is small, $\sin^{d-2}(\theta)$ is relatively flat
• When $d \to \infty$, $\sin^{d-2}(\theta)$ becomes sharply peaked at: $$\theta = \frac{\pi}{2}$$

This leads to: Most of the mass concentrates near the equator orthogonal to any fixed direction.

More formally, for any fixed $\epsilon > 0$:

This is a manifestation of the concentration of measure phenomenon on positively curved spaces.

The Geometric Interpretation could be described as “Equatorial Collapse”. In Euclidean space, samples concentrate in a thin outer shell. On the sphere, samples instead concentrate in a thin equatorial band.

This can be viewed as a curvature-induced redistribution of measure.

Angular Distance in Positive Curvature

Let:

Then their inner product satisfies:

Thus:

So we have the conclusion:

Two random points on a high-dimensional sphere are almost always orthogonal.

This implies:

Distance Concentration under Geodesic Metric

Unlike Euclidean distance, the natural metric on $\mathbb{S}^{d-1}$ is the geodesic distance:

Since:

• $\langle x, y \rangle \to 0$
• fluctuations are $O(1/\sqrt{d})$

we have:

Thus, Geodesic distances also concentrate → almost all pairs of points are at distance $\frac{\pi}{2}$.

Curvature Amplifies Concentration

An important insight is: Positive curvature does not eliminate the curse of dimensionality — it reshapes and often strengthens it.

Why?

• The sphere is compact → no radial dispersion
• Curvature forces geodesics to “bend back”
• Volume grows slower than Euclidean space

As a result: Samples are even more tightly concentrated. Angular discrimination becomes harder. nd most directions become indistinguishable

In positively curved spaces: Measure concentrates on equatorial regions, and angles concentrate around $\frac{\pi}{2}$, geodesic distances become nearly constant. The curse of dimensionality persists, but manifests as angular collapse instead of radial shell concentration.