Sparse approximation and recovery by greedy algorithms in Banach spaces

TL;DR

This paper extends the analysis of greedy algorithms for sparse approximation from Hilbert to Banach spaces, proving Lebesgue-type inequalities for the Weak Chebyshev Greedy Algorithm and demonstrating its near-optimal performance for trigonometric systems in certain Lp spaces.

Contribution

It introduces Lebesgue-type inequalities for WCGA in Banach spaces, a significant generalization from previous Hilbert space results, and applies these to bases like the trigonometric system.

Findings

WCGA provides near-optimal sparse approximation in Lp spaces for 2 ≤ p < ∞.

Theoretical bounds are established for greedy algorithms in Banach spaces.

Results extend sparse approximation theory beyond Hilbert spaces.

Abstract

We study sparse approximation by greedy algorithms. We prove the Lebesgue-type inequalities for the Weak Chebyshev Greedy Algorithm (WCGA), a generalization of the Weak Orthogonal Matching Pursuit to the case of a Banach space. The main novelty of these results is a Banach space setting instead of a Hilbert space setting. The results are proved for redundant dictionaries satisfying certain conditions. Then we apply these general results to the case of bases. In particular, we prove that the WCGA provides almost optimal sparse approximation for the trigonometric system in $L_p$, $2\le p<\infty$.