Theoretical and experimental analysis of a randomized algorithm for Sparse Fourier transform analysis

We analyze a sublinear RAℓSFA (randomized algorithm for Sparse Fourier analysis) that finds a near-optimal B-term Sparse representation R for a given discrete signal S of length N, in time and space poly ( B , log ( N ) ) , following the approach given in [A.C. Gilbert, S. Guha, P. Indyk, S. Muthukrishnan, M. Strauss, Near-Optimal Sparse Fourier Representations via Sampling, STOC, 2002]. Its time cost poly ( log ( N ) ) should be compared with the superlinear Ω ( N log N ) time requirement of the Fast Fourier Transform (FFT). A straightforward implementation of the RAℓSFA, as presented in the theoretical paper [A.C. Gilbert, S. Guha, P. Indyk, S. Muthukrishnan, M. Strauss, Near-Optimal Sparse Fourier Representations via Sampling, STOC, 2002], turns out to be very slow in practice. Our main result is a greatly improved and practical RAℓSFA. We introduce several new ideas and techniques that speed up the algorithm. Both rigorous and heuristic arguments for parameter choices are presented. Our RAℓSFA constructs, with probability at least 1 - δ , a near-optimal B-term representation R in time poly ( B ) log ( N ) log ( 1 / δ ) / ϵ 2 log ( M ) such that ∥ S - R ∥ 2 2 ⩽ ( 1 + ϵ ) ∥ S - R opt ∥ 2 2 . Furthermore, this RAℓSFA implementation already beats the FFTW for not unreasonably large N. We extend the algorithm to higher dimensional cases both theoretically and numerically. The crossover point lies at N ≃ 70 , 000 in one dimension, and at N ≃ 900 for data on a N × N grid in two dimensions for small B signals where there is noise.