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

Average Case Analysis of Multichannel Sparse Recovery Using Convex Relaxation