A sparse multidimensional FFT for real positive vectors

TL;DR

This paper introduces a randomized sparse multidimensional FFT algorithm for real positive vectors that is efficient, stable, and has a very low failure probability, suitable for high-dimensional data with few nonzeros.

Contribution

The paper proposes a novel sparse multidimensional FFT algorithm that operates efficiently in any fixed dimension with provable stability and low failure probability.

Findings

Requires O(R log R log N) samples

Runs in O(R log^2 R log N) time

Stable to low-level noise

Abstract

We present a sparse multidimensional FFT (sMFFT) randomized algorithm for real positive vectors. The algorithm works in any fixed dimension, requires (O(R log(R) log(N)) ) samples and runs in O( R log^2(R) log(N)) complexity (where N is the total size of the vector in d dimensions and R is the number of nonzeros). It is stable to low-level noise and exhibits an exponentially small probability of failure.