Entropy numbers of finite-dimensional embeddings

TL;DR

This paper surveys the known results on entropy numbers of finite-dimensional embeddings between ℓp and ℓq spaces, highlighting the techniques used in their proofs and providing a comprehensive overview.

Contribution

It offers a self-contained presentation of the entropy numbers for finite-dimensional embeddings and reviews various proof techniques used in the literature.

Findings

Entropy numbers for ℓp to ℓq embeddings are well characterized.

Multiple techniques have been developed to estimate these entropy numbers.

The survey consolidates existing results and methods in a unified framework.

Abstract

Entropy numbers and covering numbers of sets and operators are well known geometric notions, which found many applications in various fields of mathematics, statistics, and computer science. Their values for finite-dimensional embeddings $id:\ell_p^n\to \ell_q^n$, $0<p,q\le\infty$, are known (up to multiplicative constants) since the pioneering work of Sch\"utt in 1984, with later improvements by Edmunds and Triebel, K\"uhn and Gu\'edon and Litvak. The aim of this survey is to give a self-contained presentation of the result and an overview of the different techniques used in its proof.