Average Case Analysis of Multichannel Sparse Recovery Using Convex Relaxation

TL;DR

This paper analyzes the average case performance of convex relaxation methods for multichannel sparse recovery, showing that under mild conditions, the probability of failure decreases exponentially with the number of channels, highlighting the superiority over single-channel methods.

Contribution

It provides the first average case analysis of convex relaxation techniques for multichannel sparse recovery, demonstrating exponential decay in failure probability and improving bounds for other methods.

Findings

Failure probability decays exponentially with number of channels

Multichannel recovery outperforms single-channel methods in most cases

Bounds for thresholding and SOMP are tightened

Abstract

In this paper, we consider recovery of jointly sparse multichannel signals from incomplete measurements. Several approaches have been developed to recover the unknown sparse vectors from the given observations, including thresholding, simultaneous orthogonal matching pursuit (SOMP), and convex relaxation based on a mixed matrix norm. Typically, worst-case analysis is carried out in order to analyze conditions under which the algorithms are able to recover any jointly sparse set of vectors. However, such an approach is not able to provide insights into why joint sparse recovery is superior to applying standard sparse reconstruction methods to each channel individually. Previous work considered an average case analysis of thresholding and SOMP by imposing a probability model on the measured signals. In this paper, our main focus is on analysis of convex relaxation techniques. In particular, we focus on the mixed l_2,1 approach to multichannel recovery. We show that under a very mild condition on the sparsity and on the dictionary characteristics, measured for example by the coherence, the probability of recovery failure decays exponentially in the number of channels. This demonstrates that most of the time, multichannel sparse recovery is indeed superior to single channel methods. Our probability bounds are valid and meaningful even for a small number of signals. Using the tools we develop to analyze the convex relaxation method, we also tighten the previous bounds for thresholding and SOMP.