The Replica-Symmetric Prediction for Compressed Sensing with Gaussian Matrices is Exact

TL;DR

This paper rigorously confirms that the replica method's predictions for mutual information and MMSE in compressed sensing with Gaussian matrices are exact, resolving a long-standing open problem.

Contribution

It provides a rigorous proof that the replica-symmetric predictions for MI and MMSE are accurate in the high-dimensional limit for Gaussian measurement matrices.

Findings

Replica predictions for MI and MMSE are exact.

The results hold under mild technical conditions.

Resolves a long-standing open problem in compressed sensing.

Abstract

This paper considers the fundamental limit of compressed sensing for i.i.d. signal distributions and i.i.d. Gaussian measurement matrices. Its main contribution is a rigorous characterization of the asymptotic mutual information (MI) and minimum mean-square error (MMSE) in this setting. Under mild technical conditions, our results show that the limiting MI and MMSE are equal to the values predicted by the replica method from statistical physics. This resolves a well-known problem that has remained open for over a decade.