Satisfying the Restricted Isometry Property with the Optimal Number of Rows and Slightly Less Randomness

Abstract

A matrix $\Phi \in {\mathbb{R}}^{Q\times N}$ satisfies the restricted isometry property if $\Vert \Phi x{\Vert }_2^2$ is approximately equal to $\Vert x{\Vert }_2^2$ for all $k$-sparse vectors $x$. We give a construction of RIP matrices with the optimal $Q=O(k\log (N/k))$ rows using $O(k\log (N/k)\log (k))$ bits of randomness. The main technical ingredient is an extension of the Hanson-Wright inequality to $\epsilon$-biased distributions.