Empirical approximation of the gaussian distribution in $\mathbb{R}^d$

TL;DR

This paper provides an empirical approximation of the Gaussian distribution in high-dimensional space, establishing near-optimal bounds and probabilities for the accuracy of the approximation across sets with complex geometry.

Contribution

It introduces a new bound for empirical Gaussian approximation in high dimensions, involving Talagrand's $\gamma_1$ functional, and demonstrates the structural rigidity of the associated random matrix images.

Findings

Bound on empirical Gaussian approximation with high probability

Optimality of the approximation bounds and probability estimates

Structural rigidity of the random matrix images of sets

Abstract

Let $G_1,\dots,G_m$ be independent copies of the standard gaussian random vector in $\mathbb{R}^d$. We show that there is an absolute constant $c$ such that for any $A \subset S^{d-1}$, with probability at least $1-2\exp(-c\Delta m)$, for every $t\in\mathbb{R}$, \[ \sup_{x \in A} \left| \frac{1}{m}\sum_{i=1}^m 1_{ \{\langle G_i,x\rangle \leq t \}} - \mathbb{P}(\langle G,x\rangle \leq t) \right| \leq \Delta + \sigma(t) \sqrt\Delta. \] Here $\sigma(t) $ is the variance of $1_{\{\langle G,x\rangle\leq t\}}$ and $\Delta\geq \Delta_0$, where $\Delta_0$ is determined by an unexpected complexity parameter of $A$ that captures the set's geometry (Talagrand's $\gamma_1$ functional). The bound, the probability estimate, and the value of $\Delta_0$ are all (almost) optimal. We use this fact to show that if $\Gamma=\sum_{i=1}^m \langle G_i,x\rangle e_i$ is the random matrix that has $G_1,\dots,G_m$ as its rows, then the structure of $\Gamma(A)=\{\Gamma x: x\in A\}$ is far more rigid and well-prescribed than was previously expected.