(Nearly) sample-optimal sparse Fourier transform

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)O(1)) signal samples in time O(k log2 n(log log n)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(n1-δ) for a constant δ > 0.