Replica Symmetric Bound for Restricted Isometry Constant

TL;DR

This paper introduces a novel method using statistical mechanics to evaluate restricted isometry constants for Gaussian matrices, providing tighter bounds and insights into their properties.

Contribution

It develops a replica symmetric approach to estimate RICs and validates the results with advanced Monte Carlo methods, offering a new perspective on RIC bounds.

Findings

The replica symmetric estimate acts as a tighter upper bound for RICs.

Exchange Monte Carlo confirms the accuracy of the theoretical bounds.

Physical insights suggest potential improvements via replica symmetry breaking.

Abstract

We develop a method for evaluating restricted isometry constants (RICs). This evaluation is reduced to the identification of the zero-points of entropy, which is defined for submatrices that are composed of columns selected from a given measurement matrix. Using the replica method developed in statistical mechanics, we assess RICs for Gaussian random matrices under the replica symmetric (RS) assumption. In order to numerically validate the adequacy of our analysis, we employ the exchange Monte Carlo (EMC) method, which has been empirically demonstrated to achieve much higher numerical accuracy than naive Monte Carlo methods. The EMC method suggests that our theoretical estimation of an RIC corresponds to an upper bound that is tighter than in preceding studies. Physical consideration indicates that our assessment of the RIC could be improved by taking into account the replica symmetry breaking.