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

Authors:
Jing Zou;Anna Gilbert;Martin Strauss;Ingrid Daubechies
Affiliations:
Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, United States;Department of Mathematics, University of Michigan, MI, United States;Departments of Mathematics and Electrical Engineering and Computer Science, University of Michigan, MI, United States;Program in Applied and Computational Mathematics and Department of Mathematics, Princeton University, Princeton, NJ 08544, United States
Venue:
Journal of Computational Physics
Year:
2006

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Abstract

We analyze a sublinear RA@?SFA (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 @W(NlogN) time requirement of the Fast Fourier Transform (FFT). A straightforward implementation of the RA@?SFA, 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 RA@?SFA. We introduce several new ideas and techniques that speed up the algorithm. Both rigorous and heuristic arguments for parameter choices are presented. Our RA@?SFA constructs, with probability at least 1-@d, a near-optimal B-term representation R in time poly(B)log(N)log(1/@d)/@e^2log(M) such that @?S-R@?"2^2=