A conditional construction of 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.

6 pages