On the entropy numbers of the mixed smoothness function classes

TL;DR

This paper investigates the entropy numbers of multivariate function classes with mixed smoothness, introducing a new approach based on nonlinear approximation and greedy algorithms to establish upper bounds.

Contribution

It develops a novel method combining greedy approximation and volume estimates to derive bounds for entropy numbers of mixed smoothness function classes.

Findings

New upper bounds for entropy numbers using greedy approximation

Effective lower bounds via volume estimates

Advances in understanding multivariate function class complexity

Abstract

Behavior of the entropy numbers of classes of multivariate functions with mixed smoothness is studied here. This problem has a long history and some fundamental problems in the area are still open. The main goal of this paper is to develop a new method of proving the upper bounds for the entropy numbers. This method is based on recent developments of nonlinear approximation, in particular, on greedy approximation. This method consists of the following two steps strategy. At the first step we obtain bounds of the best m-term approximations with respect to a dictionary. At the second step we use general inequalities relating the entropy numbers to the best m-term approximations. For the lower bounds we use the volume estimates method, which is a well known powerful method for proving the lower bounds for the entropy numbers. It was used in a number of previous papers.