Multipath least squares algorithm and analysis

Abstract

One important task in signal processing is to construct effective algorithms to reconstruct sparse signals from an underdetermined system of linear equations. In this paper, we propose a new sparse recovery algorithm called multipath least squares (MLS), which investigates multiple promising candidates per step and parallels the multipath matching pursuit (MMP) algorithm in this aspect. The performance of the MLS algorithm is evaluated through the ability of signal recovery. Specifically, a recovery guarantee based on the restricted isometry property (RIP) is established for MLS that ensures its exact recovery of any K-sparse signal x from the measurements y=Ax. It is also shown that this sufficient condition is nearly sharp by providing a counterexample such that the algorithm may fail to recover some K-sparse signal. Moreover, the recovery guarantee of the MLS algorithm is also provided for the case of noisy measurements. Finally, numerical experiments are conducted to demonstrate the validity and priority of the proposed algorithm.