Conditional Central Limit Theorems for Gaussian Projections

TL;DR

This paper establishes bounds on how closely high-dimensional Gaussian projections of random vectors resemble a Gaussian distribution, with implications for random linear estimation and compressed sensing.

Contribution

It provides explicit bounds on the deviation between the projected distribution and a Gaussian, conditioned on the projection matrix, using Wasserstein distance and relative entropy.

Findings

Bounds depend on the number of projections and vector properties

Uses Talagrand's transportation inequality and mutual information inequalities

Applicable to compressed sensing and linear estimation

Abstract

This paper addresses the question of when projections of a high-dimensional random vector are approximately Gaussian. This problem has been studied previously in the context of high-dimensional data analysis, where the focus is on low-dimensional projections of high-dimensional point clouds. The focus of this paper is on the typical behavior when the projections are generated by an i.i.d. Gaussian projection matrix. The main results are bounds on the deviation between the conditional distribution of the projections and a Gaussian approximation, where the conditioning is on the projection matrix. The bounds are given in terms of the quadratic Wasserstein distance and relative entropy and are stated explicitly as a function of the number of projections and certain key properties of the random vector. The proof uses Talagrand's transportation inequality and a general integral-moment inequality for mutual information. Applications to random linear estimation and compressed sensing are discussed.