On application of LASSO for sparse support recovery with imperfect correlation awareness

In this paper, the problem of identifying the common sparsity support of multiple measurement vectors (MMV) is considered. The model is given by y[n] = Ax_s[n], 1 ≤ n ≤ L where {y[n]}_n=1^L 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. This result was shown assuming the knowledge that the nonzero elements are perfectly uncorrelated and that we have perfect estimates for the data correlation matrix, (the latter is true in the limit as L → ∞). In this paper, we formulate the problem of support recovery in the non ideal setting, i.e., when the correlation matrix is estimated with finite L. The resulting support recovery problem which explicitly utilizes the correlation knowledge, can be formulated as a LASSO. The performance of such “correlation aware” LASSO is analyzed by providing lower bounds on the probability of successful recovery as a function of the number L of measurement vectors. Numerical results are also provided to demonstrate the superior performance of the proposed correlation aware framework over conventional MMV techniques under identical conditions.