On the benefits of the block-sparsity structure in sparse signal recovery

We study the problem of support recovery of block-sparse signals, where nonzero entries occur in clusters, via random noisy measurements. By drawing analogy between the problem of block-sparse signal recovery and the problem of communication over Gaussian multi-input and single-output multiple access channel, we derive the sufficient and necessary condition under which exact support recovery is possible. Based on the results, we show that block-sparse signals can reduce the number of measurements required for exact support recovery, by at least `1/(block size)´, compared to conventional or scalar-sparse signals. The minimum gain is guaranteed by increased signal to noise power ratio (SNR) and reduced effective number of entries (i.e., not individual elements but blocks) that are dominant at low SNR and at high SNR, respectively. When the correlation between the elements of each nonzero block is low, a larger gain than `1/(block size)´ is expected due to, so called, diversity effect, especially in the moderate and low SNR regime.