Average Case Analysis of Multichannel Basis Pursuit

We consider the recovery of jointly sparse multichannel signals from incomplete measurements using convex relaxation methods. Worst case analysis is not able to provide insights into why joint sparse recovery is superior to applying standard sparse reconstruction methods to each channel individually. Therefore, we analyze an average case by imposing a probability model on the measured signals. 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.