Discrepancy, chaining and subgaussian processes

TL;DR

This paper investigates the behavior of coordinate projections of subgaussian function classes, showing that the minimal discrepancy over signs is smaller than expected when the associated Gaussian process is continuous, with implications for understanding the structure of such classes.

Contribution

It establishes a bound on the discrepancy of subsets in relation to Gaussian processes and provides new structural insights into coordinate projections of subgaussian classes.

Findings

Discrepancy bounds depend on Gaussian process continuity.

Coordinate projections exhibit smaller discrepancy than expectation.

Structural properties of subgaussian classes are quantitatively characterized.

Abstract

We show that for a typical coordinate projection of a subgaussian class of functions, the infimum over signs $\inf_{(\epsilon_i)}{\sup_{f\in F}}|{\sum_{i=1}^k\epsilon_i}f(X_i)|$ is asymptotically smaller than the expectation over signs as a function of the dimension $k$, if the canonical Gaussian process indexed by $F$ is continuous. To that end, we establish a bound on the discrepancy of an arbitrary subset of $\mathbb {R}^k$ using properties of the canonical Gaussian process the set indexes, and then obtain quantitative structural information on a typical coordinate projection of a subgaussian class.