The Vanishing Neighbor aka The Curse of Dimensionality
Quantifying Distance Concentration in $\mathbb{R}^d$
Task Description: Computing distances of vectors in high-dimensions is critical in machine learning. However, distances in very high dimensions are not very useful. In this task, the influence of dimensions on distinguishability of pairs of points is illustrated. At very high dimensions, all the points appear to be equally apart, and makes distance based analyses meaningless.
In low dimensions, $\sigma/\mu$ is high (wide distribution). In high dimensions, $\sigma/\mu \to 0$.
This "crushing" of the distribution into the Red Bins means the relative difference between the nearest and farthest point becomes negligible.
Distance between random points $x, y \sim \mathcal{U}(0,1)^d$