A reliable support detection is essential for a greedy algorithm toreconstruct a sparse signal accurately from compressed and noisy measurements.This paper proposes a novel support detection method for greedy algorithms,which is referred to as "\textit{maximum a posteriori (MAP) supportdetection}". Unlike existing support detection methods that identify supportindices with the largest correlation value in magnitude per iteration, theproposed method selects them with the largest likelihood ratios computed underthe true and null support hypotheses by simultaneously exploiting thedistributions of sensing matrix, sparse signal, and noise. Leveraging thistechnique, MAP-Matching Pursuit (MAP-MP) is first presented to show theadvantages of exploiting the proposed support detection method, and asufficient condition for perfect signal recovery is derived for the case whenthe sparse signal is binary. Subsequently, a set of iterative greedyalgorithms, called MAP-generalized Orthogonal Matching Pursuit (MAP-gOMP),MAP-Compressive Sampling Matching Pursuit (MAP-CoSaMP), and MAP-SubspacePursuit (MAP-SP) are presented to demonstrate the applicability of the proposedsupport detection method to existing greedy algorithms. From empirical results,it is shown that the proposed greedy algorithms with highly reliable supportdetection can be better, faster, and easier to implement than basis pursuit vialinear programming.
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