On the Paley RIP and Paley graph extractor
Shohei Satake
TL;DR
The paper builds a bridge between two seemingly unrelated conjectures: RIP-based square-root bottleneck phenomena in Paley-type matrices and half-barrier min-entropy extraction via 2-source extractors. By focusing on the Paley ETF $\Phi_p$ and Paley graph extractor Ext$_p$ derived from quadratic residues in $\mathbb{F}_p$, it shows that a (hypothetical) RIP-breaking performance of $\Phi_p$ would force Ext$_p$ to break the half barrier with negligible error. This conditional link implies that a conjectured RIP behavior for Paley constructions would entail improvements in explicit 2-source extractors, aligning with long-standing conjectures about Paley graphs and additive combinatorics. The work also outlines a program whereby RIP-constant conjectures would yield concrete consequences for Ext$_p$ and the clique structure of Paley graphs, strengthening the expected interplay between pseudorandomness, compressed sensing, and randomness extraction.
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.