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Sparsity Pattern Recovery in Bernoulli-Gaussian Signal Model

In compressive sensing, sparse signals are recovered from underdetermined noisy linear observations. One of the interesting problems which attracted a lot of attention in recent times is the support recovery or sparsity pattern recovery problem. The aim is to identify the non-zero elements in the original sparse signal. In this article we consider the sparsity pattern recovery problem under a probabilistic signal model where the sparse support follows a Bernoulli distribution and the signal restricted to this support follows a Gaussian distribution. We show that the energy in the original signal restricted to the missed support of the MAP estimate is bounded above and this bound is of the order of energy in the projection of the noise signal to the subspace spanned by the active coefficients. We also derive sufficient conditions for no misdetection and no false alarm in support recovery.