Asymptotic Analysis of MAP Estimation via the Replica Method and Applications to Compressed Sensing

TL;DR

This paper uses the replica method to analyze the asymptotic behavior of MAP estimators in compressed sensing, providing a decoupled scalar estimation framework and predicting performance metrics.

Contribution

It applies the replica method under replica symmetry to derive a decoupled scalar MAP estimation model for large random systems in compressed sensing.

Findings

Predicts asymptotic behavior of MAP estimators in compressed sensing

Provides scalar estimators for basis pursuit, lasso, and zero norm-regularized estimation

Enables precise prediction of mean-squared error and recovery probability

Abstract

The replica method is a non-rigorous but well-known technique from statistical physics used in the asymptotic analysis of large, random, nonlinear problems. This paper applies the replica method, under the assumption of replica symmetry, to study estimators that are maximum a posteriori (MAP) under a postulated prior distribution. It is shown that with random linear measurements and Gaussian noise, the replica-symmetric prediction of the asymptotic behavior of the postulated MAP estimate of an n-dimensional vector "decouples" as n scalar postulated MAP estimators. The result is based on applying a hardening argument to the replica analysis of postulated posterior mean estimators of Tanaka and of Guo and Verdu. The replica-symmetric postulated MAP analysis can be readily applied to many estimators used in compressed sensing, including basis pursuit, lasso, linear estimation with thresholding, and zero norm-regularized estimation. In the case of lasso estimation the scalar estimator reduces to a soft-thresholding operator, and for zero norm-regularized estimation it reduces to a hard-threshold. Among other benefits, the replica method provides a computationally-tractable method for precisely predicting various performance metrics including mean-squared error and sparsity pattern recovery probability.