Entropy-based convergence rates of greedy algorithms

TL;DR

This paper introduces entropy-based convergence estimates for greedy algorithms, providing sharper bounds and direct comparisons to metric entropy, enhancing understanding of reduced basis methods and nonlinear approximation.

Contribution

It offers the first entropy-based convergence analysis for greedy algorithms, improving classical width-based bounds and providing explicit constants.

Findings

Entropy-based estimates are sharp and improve classical bounds.

Provides explicit constants in convergence estimates.

Establishes direct links between algorithm error and metric entropy.

Abstract

We present convergence estimates of two types of greedy algorithms in terms of the metric entropy of underlying compact sets. In the first part, we measure the error of a standard greedy reduced basis method for parametric PDEs by the metric entropy of the solution manifold in Banach spaces. This contrasts with the classical analysis based on the Kolmogorov n-widths and enables us to obtain direct comparisons between the greedy algorithm error and the entropy numbers, where the multiplicative constants are explicit and simple. The entropy-based convergence estimate is sharp and improves upon the classical width-based analysis of reduced basis methods for elliptic model problems. In the second part, we derive a novel and simple convergence analysis of the classical orthogonal greedy algorithm for nonlinear dictionary approximation using the metric entropy of the symmetric convex hull of the dictionary. This also improves upon existing results by giving a direct comparison between the algorithm error and the metric entropy.