Combinatorics of random processes and sections of convex bodies

TL;DR

This paper establishes sharp bounds on the metric entropy of sets and function classes, solving key conjectures in empirical processes and connecting combinatorial dimensions with entropy, with implications for convex geometry and operator theory.

Contribution

It provides a sharp combinatorial bound for metric entropy, resolving two fundamental conjectures in empirical processes and linking entropy to combinatorial dimension.

Findings

Proves a class satisfies the CLT if its combinatorial dimension's square root is integrable.

Shows uniform entropy is equivalent to combinatorial dimension under minimal regularity.

Constructs bounded coordinate sections of convex bodies, impacting operator theory.

Abstract

We find a sharp combinatorial bound for the metric entropy of sets in R^n and general classes of functions. This solves two basic combinatorial conjectures on the empirical processes. 1. A class of functions satisfies the uniform Central Limit Theorem if the square root of its combinatorial dimension is integrable. 2. The uniform entropy is equivalent to the combinatorial dimension under minimal regularity. Our method also constructs a nicely bounded coordinate section of a symmetric convex body in R^n. In the operator theory, this essentially proves for all normed spaces the restricted invertibility principle of Bourgain and Tzafriri.