Sparse approximation by greedy algorithms

TL;DR

This survey reviews recent advances in constructive sparse approximation using greedy algorithms, covering Lebesgue inequalities, trigonometric systems, and tensor product dictionaries, with detailed proofs and connections to compressed sensing.

Contribution

It provides a comprehensive overview of recent results in greedy-based sparse approximation, including new inequalities and constructive methods for various dictionary types.

Findings

Lebesgue-type inequalities for greedy algorithms

Constructive sparse approximation methods for trigonometric systems

Sparse approximation techniques for tensor product dictionaries

Abstract

It is a survey on recent results in constructive sparse approximation. Three directions are discussed here: (1) Lebesgue-type inequalities for greedy algorithms with respect to a special class of dictionaries, (2) constructive sparse approximation with respect to the trigonometric system, (3) sparse approximation with respect to dictionaries with tensor product structure. In all three cases constructive ways are provided for sparse approximation. The technique used is based on fundamental results from the theory of greedy approximation. In particular, results in the direction (1) are based on deep methods developed recently in compressed sensing. We present some of these results with detailed proofs.