The Restricted Isometry Property of Subsampled Fourier Matrices

A matrix A ∈ mathbbCq × N satisfies the restricted isometry property of order k with constant varepsilon if it preserves the ℓ2 norm of all k-sparse vectors up to a factor of 1± varepsilon. We prove that a matrix A obtained by randomly sampling q = O(k · log2 k · log N) rows from an N × N Fourier matrix satisfies the restricted isometry property of order k with a fixed varepsilon with high probability. This improves on Rudelson and Vershynin (Comm. Pure Appl. Math., 2008), its subsequent improvements, and Bourgain (GAFA Seminar Notes, 2014).