Dvoretzky type theorems for subgaussian coordinate projections

TL;DR

This paper extends Dvoretzky's theorem to subgaussian function classes, demonstrating that typical coordinate projections of such classes exhibit nearly Euclidean structure, generalizing linear random projections in geometric analysis.

Contribution

It introduces a Dvoretzky type theorem for subgaussian classes of functions, broadening the scope of geometric analysis to nonlinear function classes.

Findings

Typical coordinate projections of subgaussian classes are nearly Euclidean.

The results generalize linear random projection properties in geometric analysis.

Provides a framework for understanding the structure of function class projections.

Abstract

Given a class of functions $F$ on a probability space $(\Omega,\mu)$, we study the structure of a typical coordinate projection of the class, defined by $\{(f(X_i))_{i=1}^N : f \in F\}$, where $X_1,...,X_N$ are independent, selected according to $\mu$. This notion of projection generalizes the standard linear random projection used in Asymptotic Geometric Analysis. We show that when $F$ is a subgaussian class of functions, a typical coordinate projection satisfies a Dvoretzky type theorem.