Correlation-aware sparse support recovery: Gaussian sources

Consider a multiple measurement vector (MMV) model given by y[n] = Axs[n]; 1 ≤ n ≤ L where equation denote the L measurement vectors, A ∈ R^M×N is the measurement matrix and x_s[n] ∈ R^N are the unknown vectors with same sparsity support denoted by the set S₀ with |S₀| = D. It has been shown in a recent paper by the authors that when the elements of x_s[n] are uncorrelated from each other, one can recover sparsity levels as high as O(M²) for suitably designed measurement matrix. The recovery is exact when support recovery algorithms are applied on the ideal correlation matrix. When we only have estimates of the correlation, it is still possible to probabilistically argue the recovery of sparsity levels (using a coherence based argument) that is much higher than that guaranteed by existing coherence based results. However the lower bound on the probability of success is found to increase rather slowly with L (as 1-C/L for some constant C > 0) without any further assumption on the distribution of the source vectors. In this paper, we demonstrate that when the source vectors belong to a Gaussian distribution with diagonal covariance matrix, it is possible to guarantee the recovery of original support with overwhelming probability. We also provide numerical simulations to demonstrate the effectiveness of the proposed strategy by comparing it with other popular MMV based methods.