Performance of Jointly Sparse Support Recovery in Compressed Sensing

The problem of jointly sparse support recovery is to determine the common support of jointly sparse signal vectors from multiple measurement vectors (MMV) related to the signals by a linear transformation. The fundamental limit of performance has been studied in terms of a so-called algebraic bound, relating the maximum recoverable sparsity level to the spark of the sensing matrix and the rank of the signal matrix. However, while the algebraic bound provides the necessary and sufficient condition for the success of joint sparse recovery, it is restricted to the noiseless case. We derive a sufficient condition for jointly sparse support recovery for the noisy case. We show that essentially the same deterministic condition as in the noiseless case suffices for perfect support recovery at finite signal-to-noise ratio (SNR). Furthermore, we perform an average case analysis of the recovery problem when the matrix of jointly sparse signal vectors has random left singular vectors, representing the case of signal vectors in general position. In this case, we provide a relaxed deterministic condition on the sensing matrix for support recovery with high probability at finite SNR. Finally, we quantify the improvements for an i.i.d. Gaussian sensing matrix.