A study of the universal threshold in the L1 recovery by statistical mechanics

TL;DR

This paper investigates the universality of the L1 recovery threshold in compressed sensing, demonstrating analytically and numerically that the threshold depends on the symmetry properties of the random matrix used.

Contribution

It provides an analytical and numerical analysis showing that the universality of the L1 recovery threshold is influenced by the symmetry of the measurement matrix, challenging previous assumptions.

Findings

Universal threshold holds for orthogonal symmetric matrices

Non-orthogonal matrices have different recovery thresholds

Numerical experiments confirm analytical predictions

Abstract

We discuss the universality of the L1 recovery threshold in compressed sensing. Previous studies in the fields of statistical mechanics and random matrix integration have shown that L1 recovery under a random matrix with orthogonal symmetry has a universal threshold. This indicates that the threshold of L1 recovery under a non-orthogonal random matrix differs from the universal one. Taking this into account, we use a simple random matrix without orthogonal symmetry, where the random entries are not independent, and show analytically that the threshold of L1 recovery for such a matrix does not coincide with the universal one. The results of an extensive numerical experiment are in good agreement with the analytical results, which validates our methodology. Though our analysis is based on replica heuristics in statistical mechanics and is not rigorous, the findings nevertheless support the fact that the universality of the threshold is strongly related to the symmetry of the random matrix.