Coherence-Based Performance Guarantees for Estimating a Sparse Vector Under Random Noise

TL;DR

This paper provides non-asymptotic, coherence-based performance guarantees for sparse vector estimation algorithms (BPDN, OMP, thresholding) under Gaussian noise, highlighting their near-oracle performance and comparative advantages.

Contribution

It offers the first coherence-based, non-asymptotic analysis of multiple sparse estimation algorithms under noise, applicable to arbitrary dictionaries.

Findings

Algorithms achieve near-oracle performance with high probability.

Different algorithms have varying performance depending on SNR levels.

Coherence of the dictionary is key to performance guarantees.

Abstract

We consider the problem of estimating a deterministic sparse vector x from underdetermined measurements Ax+w, where w represents white Gaussian noise and A is a given deterministic dictionary. We analyze the performance of three sparse estimation algorithms: basis pursuit denoising (BPDN), orthogonal matching pursuit (OMP), and thresholding. These algorithms are shown to achieve near-oracle performance with high probability, assuming that x is sufficiently sparse. Our results are non-asymptotic and are based only on the coherence of A, so that they are applicable to arbitrary dictionaries. Differences in the precise conditions required for the performance guarantees of each algorithm are manifested in the observed performance at high and low signal-to-noise ratios. This provides insight on the advantages and drawbacks of convex relaxation techniques such as BPDN as opposed to greedy approaches such as OMP and thresholding.