MCMC Sampling-Based Randomized Likelihood Decoding for Sparse Recovery

We investigate whether randomized likelihood (RL) decoding based on sampling techniques can completely replace the compressed sensing (CS) process for sparse recovery. For a Gaussian signal model, we propose a novel iterative Markov chain Monte Carlo (MCMC) sampling-based RL decoding method tailored to the attributes of sparse recovery, termed MCMC-RLD-SR. The proposed iterative MCMC-RLD-SR algorithm incorporates two stages, i.e., rough estimation and fine estimation. The rough estimation is a process of figuring out support candidates for the sparse signal via the Metropolis-Hastings (MH) sampling method, which prevents a nonconvergence issue inherent in the CS problem when applying sampling. The fine estimation is a process of acquiring an estimate of the sparse signal through the Gibbs sampling method based on the support candidates from the rough estimation stage. We prove that the proposed algorithm converges by favor of the proposed iterative two-stage sampling structure, and analyze the signal recovery error by the proposed algorithm. Our analysis and simulation results show that the proposed MCMC-RLD-SR algorithm can effectively solve CS problems with much less computational complexity than conventional CS algorithms. Furthermore, even when the signal is not sparse, the proposed algorithm is shown to achieve a reliable signal recovery performance.