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

A matrix Φ∈RQ×N satisfies the restricted isometry property if ‖Φx‖22 is approximately equal to ‖x‖22 for all k-sparse vectors x. We give a construction of RIP matrices with the optimal Q=O(klog⁡(N/k)) rows using O(klog⁡(N/k)log⁡(k)) bits of randomness. The main technical ingredient is an extension of the Hanson-Wright inequality to ε-biased distributions. •We analyze RIP matrices with the optimal number of rows.•We show that the amount of randomness needed is less than other known constructions.•The proof follows from a generalization of the Hanson-Wright inequality.