Derandomizing restricted isometries via the Legendre symbol

TL;DR

This paper introduces a method to derandomize restricted isometry matrices in compressed sensing using the Legendre symbol, reducing randomness and potentially eliminating it altogether, with broad applicability to small-bias spaces.

Contribution

It presents a novel derandomization technique for RIP matrices leveraging the Legendre symbol, generalizable to small-bias sample spaces, and conjectures minimal randomness requirements.

Findings

Reduces random bits in RIP matrix construction

Generalizes to small-bias sample spaces

Conjectures no randomness needed for Legendre symbol-based construction

Abstract

The restricted isometry property (RIP) is an important matrix condition in compressed sensing, but the best matrix constructions to date use randomness. This paper leverages pseudorandom properties of the Legendre symbol to reduce the number of random bits in an RIP matrix with Bernoulli entries. In this regard, the Legendre symbol is not special---our main result naturally generalizes to any small-bias sample space. We also conjecture that no random bits are necessary for our Legendre symbol--based construction.