The Restricted Isometry Property of Subsampled Fourier Matrices

A matrix A ∈ Cq×N satisfies the restricted isometry property of order k with constant ε if it preserves the ℓ2 norm of all k-sparse vectors up to a factor of 1 ± ε. We prove that a matrix A obtained by randomly sampling q = O(k · log2k · log N) rows from an N ×N Fourier matrix satisfies the restricted isometry property of order k with a fixed ε with high probability. This improves on Rudelson and Vershynin (Comm. Pure Appl. Math., 2008), its subsequent improvements, and Bourgain (GAFA Seminar Notes, 2014).