Satisfying the restricted isometry property with the optimal number of rows and slightly less randomness

Abstract

A matrix $\Phi \in R^{Q\times N}$ 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 $Q=O(k\log (N/k\left)\right)$ rows using $O(k\log (N/k)\log (k\left)\right)$ bits of randomness. The main technical ingredient is an extension of the Hanson-Wright inequality to ε-biased distributions.