Performance analysis of support recovery with joint sparsity constraints

In this paper, we analyze the performance of estimating the common support for jointly sparse signals based on their projections onto lower-dimensional space. We formulate support recovery as a multiple-hypothesis testing problem and derive both upper and lower bounds on the probability of error for general measurement matrices, by using Chernoff bound and Fano's inequality, respectively. When applied to Gaussian measurement ensembles, these bounds give necessary and sufficient conditions to guarantee a vanishing probability of error for majority realizations of the measurement matrix. Our results offer surprising insights into sparse signal reconstruction based on their projections. For example, as far as support recovery is concerned, the well-known bound in compressive sensing is generally not sufficient if the Gaussian ensemble is used. Our study provides an alternative performance measure, one that is natural and important in practice, for signal recovery in compressive sensing as well as other application areas taking advantage of signal sparsity.