Sparse approximation and recovery by greedy algorithms

TL;DR

This paper analyzes the effectiveness of greedy algorithms, particularly Orthogonal Matching Pursuit, for sparse signal recovery, providing probabilistic guarantees and extending results to Banach spaces.

Contribution

It proves high-probability exact recovery of sparse signals with OMP and extends Lebesgue-type inequalities to Banach spaces, broadening theoretical understanding.

Findings

OMP achieves near-optimal recovery within a small number of iterations.

Lebesgue-type inequalities are established for the Weak Chebyshev Greedy Algorithm in Banach spaces.

Results extend known bounds from Hilbert spaces to Banach spaces.

Abstract

We study sparse approximation by greedy algorithms. Our contribution is two-fold. First, we prove exact recovery with high probability of random $K$-sparse signals within $\lceil K(1+\e)\rceil$ iterations of the Orthogonal Matching Pursuit (OMP). This result shows that in a probabilistic sense the OMP is almost optimal for exact recovery. Second, we prove the Lebesgue-type inequalities for the Weak Chebyshev Greedy Algorithm, 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. However, even in the case of a Hilbert space our results add some new elements to known results on the Lebesque-type inequalities for the RIP dictionaries. Our technique is a development of the recent technique created by Zhang.