# Replica Symmetry Breaking in Compressive Sensing

## TL;DR

This paper analyzes the asymptotic distortion in noisy compressive sensing systems using a statistical mechanical approach, incorporating replica symmetry breaking to improve performance predictions at high compression rates.

## Contribution

It introduces a comprehensive replica analysis including RSB for compressive sensing, extending beyond the traditional RS assumption to better predict system performance.

## Key findings

- RS fails at large compression rates for zero-norm penalty
- One-step RSB provides accurate predictions in a broader regime
- RSB captures the impact of symmetry breaking on reconstruction performance

## Abstract

For noisy compressive sensing systems, the asymptotic distortion with respect to an arbitrary distortion function is determined when a general class of least-square based reconstruction schemes is employed. The sampling matrix is considered to belong to a large ensemble of random matrices including i.i.d. and projector matrices, and the source vector is assumed to be i.i.d. with a desired distribution. We take a statistical mechanical approach by representing the asymptotic distortion as a macroscopic parameter of a spin glass and employing the replica method for the large-system analysis. In contrast to earlier studies, we evaluate the general replica ansatz which includes the RS ansatz as well as RSB. The generality of the solution enables us to study the impact of symmetry breaking. Our numerical investigations depict that for the reconstruction scheme with the "zero-norm" penalty function, the RS fails to predict the asymptotic distortion for relatively large compression rates; however, the one-step RSB ansatz gives a valid prediction of the performance within a larger regime of compression rates.