Greedy Algorithms for Sparse and Positive Signal Recovery Based on Bit-Wise MAP Detection

We propose a novel greedy algorithm to recover a sparse signal from a small number of noisy measurements. In the proposed method, a new support index is identified for each iteration based on bit-wise maximum a posteriori (B-MAP) detection. This is an optimal in the sense of recovering one of the remaining support indices, provided that all the detected indices during the previous iterations are correct. Despite its optimality, it is unpractical due to an expensive complexity as the computation of the maximization metric (i.e., a posteriori probability of each remaining support) requires the marginalization of high-dimensional sparse vector. We address this problem by presenting a good proxy (named B-MAP proxy) of the maximization metric. The proposed B-MAP proxy is accurate enough to accomplish our purpose (i.e., finding a maximum index during iterations) and simply evaluated via simple vector correlations as in conventional orthogonal-matching-pursuit (OMP). We verify that, when non-zero values of a sparse signal follow a probability distribution with non-zero mean, the proposed B-MAP attains a higher recovery accuracy than the existing methods as OMP and MAP-OMP, while having the same complexity. Subsequently, advanced greedy algorithms, based on B-MAP proxy, are constructed using the idea of compressive-sampling-matching-pursuit (CoSaMP) and subspace-pursuit (SP). Simulations demonstrate that B-MAP can outperform OMP and MAP-OMP under the frameworks of the advanced greedy algorithms. We remark that the performance gains of the proposed algorithms are larger as the means of non-zero values of a random sparse signal become far from zero.