(Nearly) sample-optimal sparse fourier transform

Abstract

We consider the problem of computing a k-sparse approximation to the discrete Fourier transform of an n-dimensional signal. Our main result is a randomized algorithm that computes such an approximation using O(k log n(log log n)[superscript O(1)]) signal samples in time O(k log[superscript 2] n(log log n)[superscript O(1)]), assuming that the entries of the signal are polynomially bounded. The sampling complexity improves over the recent bound of O(k log n log(n/k)) given in [15], and matches the lower bound of Ω(k log(n/k)/log log n) from the same paper up to poly(log log n) factors when k = O(n[superscript 1-δ]) for a constant δ > 0.