Nearly Optimal Sparse Fourier Transform

TL;DR

This paper introduces nearly optimal algorithms for computing sparse Fourier transforms faster than the traditional FFT, with proven lower bounds on sampling complexity for general signals.

Contribution

It presents the first sub-linear time randomized algorithms for sparse Fourier transform computation with optimal or near-optimal performance.

Findings

O(k log n) time algorithm for exactly k-sparse signals

O(k log n log(n/k)) time algorithm for general signals

Lower bound of (k log(n/k)/ log log n) samples for any algorithm

Abstract

We consider the problem of computing the k-sparse approximation to the discrete Fourier transform of an n-dimensional signal. We show: * An O(k log n)-time randomized algorithm for the case where the input signal has at most k non-zero Fourier coefficients, and * An O(k log n log(n/k))-time randomized algorithm for general input signals. Both algorithms achieve o(n log n) time, and thus improve over the Fast Fourier Transform, for any k = o(n). They are the first known algorithms that satisfy this property. Also, if one assumes that the Fast Fourier Transform is optimal, the algorithm for the exactly k-sparse case is optimal for any k = n^{\Omega(1)}. We complement our algorithmic results by showing that any algorithm for computing the sparse Fourier transform of a general signal must use at least \Omega(k log(n/k)/ log log n) signal samples, even if it is allowed to perform adaptive sampling.