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Minimum mean square error (MMSE) estimation of block sparse signals fromnoisy linear measurements is considered. Unlike in the standard compressivesensing setup where the non-zero entries of the signal are independently anduniformly distributed across the vector of interest, the information bearingcomponents appear here in large mutually dependent clusters. Using the replicamethod from statistical physics, we derive a simple closed-form solution forthe MMSE obtained by the optimum estimator. We show that the MMSE is a versionof the Tse-Hanly formula with system load and MSE scaled by parameters thatdepend on the sparsity pattern of the source. It turns out that this is equalto the MSE obtained by a genie-aided MMSE estimator which is informed inadvance about the exact locations of the non-zero blocks. The asymptoticresults obtained by the non-rigorous replica method are found to have anexcellent agreement with finite sized numerical simulations.

Analysis of MMSE Estimation for Compressive Sensing of Block Sparse Signals