Fast Recursive Greedy Methods for Sparse Signal Recovery

Theory and algorithms of greedy methods for sparse signal recovery using₀-minimization are developed. The theoretical analysis shows that₀-minimization is more suitable for sparse signal recovery and that the greedy method based on active set updating can solve₀-minimization problems under weak conditions. A fast recursive algorithm RASU is given to greedily update the active set with a given sparsity level, and an adaptive strategy is proposed to escape from the local minima achieved by RASU. RASU is further applied to minimize the residual. The algorithm FRASM repeats this procedure, using multiple starting points to enhance the efficiency. A fast algorithm SHTP is proposed to provide the multiple choices for FRASM, and itself is also an efficient algorithm solving the sparse problem. By combining an adaptive updating rule to estimate the sparsity of the sparsest signal, the FRASM is adopted to minimize the sparsity level subject to a residual constraint, yielding the algorithm SASSP. Numerical experiments demonstrate the superior performance of FRASM and SASSP compared to other sparse algorithms.