Compressed Sensing Performance Analysis via Replica Method Using Bayesian Framework

Abstract

Compressive sensing (CS) is a new methodology to capture signals at lower rate than the Nyquist sampling rate when the signals are sparse or sparse in some domain. Studying the performance of such novel paradigm is an interesting subject. In this paper, the performance of CS estimators is analyzed using tools from statistical mechanics, especially called replica method via Bayesian framework. This method has been used to analyze communication systems like Code Division Multiple Access (CDMA) and multiple input multiple output (MIMO) systems with large size. Replica analysis, partially proved to be rigorous, is an efficient tool to analyze large systems in general. Specifically, we analyze the performance of some of the estimators used in CS like LASSO (the Least Absolute Shrinkage and Selection Operator) estimator and Zero-Norm regularizing estimator as a special case of maximum a posterior (MAP) estimator by using Bayesian framework to connect the CS estimators and replica method. We use both replica symmetric (RS) ansatz and one-step replica symmetry breaking (1RSB) ansatz, clamming the latter is efficient when the problem is not convex. This work is more analytical in its form. It is deferred for next step to focus on the numerical results.