On the Paley RIP and Paley graph extractor

TL;DR

This paper links the construction of explicit RIP matrices in compressed sensing to the development of 2-source extractors in theoretical computer science, focusing on Paley-based structures, and suggests that progress in one area implies progress in the other.

Contribution

It establishes a novel connection between Paley ETF matrices and Paley graph extractors, implying that breaking the square-root bottleneck in RIP matrices would also break the half barrier in 2-source extractors.

Findings

If Paley ETF breaks the square-root bottleneck, then Paley graph extractor breaks the half barrier.

Provides new intuition supporting the conjecture that Paley ETF breaks the square-root bottleneck.

Links two important open problems in compressed sensing and theoretical computer science.

Abstract

Constructing explicit RIP matrices is an open problem in compressed sensing theory. In particular, it is quite challenging to construct explicit RIP matrices that break the square-root bottleneck. On the other hand, providing explicit $2$-source extractors is a fundamental problem in theoretical computer science, cryptography and combinatorics. Nowadays, there are only a few known constructions for explicit $2$-source extractors (with negligible errors) that break the half barrier for min-entropy. In this paper, we establish a new connection between RIP matrices breaking the square-root bottleneck and $2$-source extractors breaking the half barrier for min-entropy. Here we focus on an RIP matrix (called the Paley ETF) and a $2$-source extractor (called the Paley graph extractor), where both are defined from quadratic residues over the finite field of odd prime order $p\equiv 1 \pmod{4}$. As a main result, we prove that if the Paley ETF breaks the square-root bottleneck, then the Paley graph extractor breaks the half barrier for min-entropy as well. Since it is widely believed that the Paley ETF breaks the square-root bottleneck, our result accordingly provides a new affirmative intuition on the conjecture for the Paley graph extractor by Benny Chor and Oded Goldreich.