Average Case Analysis of Multichannel Sparse Recovery Using Convex Relaxation

This paper considers 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. Here, the main focus is on analysis of convex relaxation techniques. In particular, the mixed
l

_2,1
approach to multichannel recovery is investigated. Under a very mild condition on the sparsity and on the dictionary characteristics, measured for example by the coherence, it is shown that 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. The probability bounds are valid and meaningful even for a small number of signals. Using the tools developed to analyze the convex relaxation technique, also previous bounds for thresholding and SOMP recovery are tightened.