Maximum-A-Posteriori Signal Recovery with Prior Information: Applications to Compressive Sensing

TL;DR

This paper analyzes the asymptotic performance of MAP estimation with prior information in signal recovery, revealing an asymmetric decoupling property and demonstrating improved measurement efficiency with optimal weighting in compressive sensing.

Contribution

It introduces a generic MAP estimator framework with prior info, studies its asymptotic behavior using the replica method, and shows how weighted zero-norm minimization enhances sparse signal recovery.

Findings

Asymmetric decoupling property in large-system limit

Optimal weighting reduces the number of measurements needed

Performance analysis via the replica method

Abstract

This paper studies the asymptotic performance of maximum-a-posteriori estimation in the presence of prior information. The problem arises in several applications such as recovery of signals with non-uniform sparsity pattern from underdetermined measurements. With prior information, the maximum-a-posteriori estimator might have asymmetric penalty. We consider a generic form of this estimator and study its performance via the replica method. Our analyses demonstrate an asymmetric form of the decoupling property in the large-system limit. Employing our results, we further investigate the performance of weighted zero-norm minimization for recovery of a non-uniform sparse signal. Our investigations illustrate that for a given distortion, the minimum number of required measurements can be significantly reduced by choosing weighting coefficients optimally.