Ellipsoid Methods for Metric Entropy Computation

TL;DR

This paper introduces a novel ellipsoid-based method for computing the metric entropy of infinite-dimensional ellipsoids, offering a unified approach that improves existing results across various analytic function classes.

Contribution

It develops a new methodology that avoids explicit coverings, providing a unified framework for metric entropy computation of diverse analytic function classes.

Findings

Improves metric entropy bounds for analytic function classes

Provides a unified framework applicable to various function spaces

Enhances understanding of infinite-dimensional ellipsoid properties

Abstract

We present a new methodology for the characterization of the metric entropy of infinite-dimensional ellipsoids with exponentially decaying semi-axes. This procedure does not rely on the explicit construction of coverings or packings and provides a unified framework for the derivation of the metric entropy of a wide variety of analytic function classes, such as periodic functions analytic on a strip, analytic functions bounded on a disk, and functions of exponential type. In each of these cases, our results improve upon the best known results in the literature.