Concentration of Measure for Radial Distributions and Consequences for Statistical Modeling

TL;DR

This paper studies how radial distributions behave in high dimensions, showing concentration phenomena similar to Gaussian distributions, and explores implications for high-dimensional statistical models like mixture modeling.

Contribution

It extends the concentration of measure results from Gaussian to general radial densities and discusses potential universality in high-dimensional mixture models.

Findings

Radial densities concentrate around a sphere as dimension increases

Convergence in distribution for radial densities under certain conditions

Implications for universality in high-dimensional Gaussian mixture models

Abstract

Motivated by problems in high-dimensional statistics such as mixture modeling for classification and clustering, we consider the behavior of radial densities as the dimension increases. We establish a form of concentration of measure, and even a convergence in distribution, under additional assumptions. This extends the well-known behavior of the normal distribution (its concentration around the sphere of radius square-root of the dimension) to other radial densities. We draw some possible consequences for statistical modeling in high-dimensions, including a possible universality property of Gaussian mixtures.