Statistical mechanical assessment of a reconstruction limit of compressed sensing: Toward theoretical analysis of correlated signals

TL;DR

This paper uses statistical mechanics to analyze the theoretical limits of reconstructing correlated signals in compressed sensing, providing insights into the critical measurement rate needed for perfect recovery.

Contribution

It introduces a replica-based scheme for assessing reconstruction limits in compressed sensing with correlated signals, specifically applied to sparse autoregressive models.

Findings

The scheme accurately predicts the critical compression rate for perfect reconstruction.

Results align well with numerical experiments on sparse autoregressive signals.

Provides a theoretical framework for correlated signal reconstruction in compressed sensing.

Abstract

We provide a scheme for exploring the reconstruction limit of compressed sensing by minimizing the general cost function under the random measurement constraints for generic correlated signal sources. Our scheme is based on the statistical mechanical replica method for dealing with random systems. As a simple but non-trivial example, we apply the scheme to a sparse autoregressive model, where the first differences in the input signals of the correlated time series are sparse, and evaluate the critical compression rate for a perfect reconstruction. The results are in good agreement with a numerical experiment for a signal reconstruction.