Compressed Sensing Performance Analysis via Replica Method using Bayesian framework

TL;DR

This paper applies the replica method from statistical mechanics within a Bayesian framework to analyze the performance of compressive sensing estimators, including LASSO and zero-norm regularization, especially in large systems.

Contribution

It introduces a Bayesian approach to connect CS estimators with the replica method, analyzing their performance using replica symmetric and one-step replica symmetry breaking ansatz.

Findings

Analytical performance analysis of CS estimators.

Use of replica symmetry breaking for non-convex problems.

Framework applicable to large-scale systems.

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. The performance of CS estimators is analyzed in this paper using tools from statistical mechanics, especially called replica method. This method has been used to analyze communication systems like Code Division Multiple Access (CDMA) and multiple input multi- ple output (MIMO) systems with large size. Replica analysis, now days rigorously proved, 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 posteriori (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.