Explicit RIP matrices: an update

TL;DR

This paper presents improved explicit constructions of RIP matrices using additive combinatorics, achieving better parameters for large dimensions and sparsity levels, which enhances compressed sensing applications.

Contribution

It provides explicit RIP matrices with improved parameters leveraging recent advances in additive combinatorics, surpassing previous constructions.

Findings

Constructed RIP matrices with better parameters for large dimensions

Achieved matrices with sparsity level $k = ext{Omega}(n^{1/2+ ext{epsilon}/4})$

Applicable to compressed sensing with improved guarantees

Abstract

Leveraging recent advances in additive combinatorics, we exhibit explicit matrices satisfying the Restricted Isometry Property with better parameters. Namely, for $\varepsilon=3.26\cdot 10^{-7}$, large $k$ and $k^{2-\varepsilon} \le N\le k^{2+\varepsilon}$, we construct $n \times N$ RIP matrices of order $k$ with $k = \Omega( n^{1/2+\varepsilon/4} )$.