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⋅logN) 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).Comment: 16 page