Improving Approximate Message Passing Recovery of Sparse Binary Vectors by Post Processing

Compressed sensing allows to recover a sparse highdimensional signal vector from fewer measurements than its dimension would suggest. Approximate Message Passing (AMP) was introduced by Donoho et al. to perform iterative recovery with low complexity. We investigate the behavior of AMP for a low number of measurements and low signal-to-noise ratio. In these cases, recovered signals exhibit many false alarms and strong amplitude deviations from the original. We propose post processing schemes that try to retain the correct detections while voiding false alarms. The best performance is obtained by Maximum Likelihood (ML) detection that is performed on a well selected subset of the recovered signal vector's support. In order to maintain feasibility, ML detection is implemented by a sphere decoder and the sparse signal vector is assumed to be binary, i.e., its few nonzero elements have value one. We show by simulation that post processing enables accurate estimation of the true signal vector and its support, despite strong noise or an insufficient number of measurements.