On Compressed Sensing Matrices Breaking the Square-Root Bottleneck

TL;DR

This paper explores deterministic matrices based on higher power residues modulo primes, demonstrating that under a conjecture, these matrices can surpass the traditional sparsity limits in compressed sensing.

Contribution

It links the RIP of specific matrices to the generalized Paley graph conjecture, suggesting a pathway to break the square-root bottleneck in compressed sensing.

Findings

Matrices based on higher power residues can have RIP beyond the square-root limit

The generalized Paley graph conjecture implies these matrices can break the bottleneck

Provides theoretical evidence connecting number theory and compressed sensing

Abstract

Compressed sensing is a celebrated framework in signal processing and has many practical applications. One of challenging problems in compressed sensing is to construct deterministic matrices having restricted isometry property (RIP). So far, there are only a few publications providing deterministic RIP matrices beating the square-root bottleneck on the sparsity level. In this paper, we investigate RIP of certain matrices defined by higher power residues modulo primes. Moreover, we prove that the widely-believed generalized Paley graph conjecture implies that these matrices have RIP breaking the square-root bottleneck.