A conditional construction of restricted isometries

TL;DR

This paper investigates the restricted isometry property of a Fourier-based matrix constructed from quadratic residues, showing it satisfies RIP under certain number-theoretic conjectures with specific sparsity bounds.

Contribution

It introduces a new construction of restricted isometries using quadratic residues and establishes RIP conditions conditioned on a folklore number theory conjecture.

Findings

Matrix satisfies RIP with sparsity $K= ext{Omega}(M^{1/2+ ext{epsilon}})$

Conditioned on a folklore conjecture in number theory

Provides a new approach to constructing restricted isometries

Abstract

We study the restricted isometry property of a matrix that is built from the discrete Fourier transform matrix by collecting rows indexed by quadratic residues. We find an $\epsilon>0$ such that, conditioned on a folklore conjecture in number theory, this matrix satisfies the restricted isometry property with sparsity parameter $K=\Omega(M^{1/2+\epsilon})$, where $M$ is the number of rows.