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Combinatorial Sublinear-Time Fourier Algorithms

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Abstract

We study the problem of estimating the best k term Fourier representation for a given frequency sparse signal (i.e., vector) A of length N≫k. More explicitly, we investigate how to deterministically identify k of the largest magnitude frequencies of \(\hat{\mathbf{A}}\) , and estimate their coefficients, in polynomial(k,log N) time. Randomized sublinear-time algorithms which have a small (controllable) probability of failure for each processed signal exist for solving this problem (Gilbert et al. in ACM STOC, pp. 152–161, 2002; Proceedings of SPIE Wavelets XI, 2005). In this paper we develop the first known deterministic sublinear-time sparse Fourier Transform algorithm which is guaranteed to produce accurate results. As an added bonus, a simple relaxation of our deterministic Fourier result leads to a new Monte Carlo Fourier algorithm with similar runtime/sampling bounds to the current best randomized Fourier method (Gilbert et al. in Proceedings of SPIE Wavelets XI, 2005). Finally, the Fourier algorithm we develop here implies a simpler optimized version of the deterministic compressed sensing method previously developed in (Iwen in Proc. of ACM-SIAM Symposium on Discrete Algorithms (SODA’08), 2008).

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Authors and Affiliations

  1. Institute for Mathematics and its Applications (IMA), University of Minnesota, Minneapolis, USA

    M. A. Iwen

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  1. M. A. Iwen
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Correspondence to M. A. Iwen.

Additional information

Communicated by Eitan Tadmor.

Results herein supersede preliminary Fourier results in [28, 29].

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Iwen, M.A. Combinatorial Sublinear-Time Fourier Algorithms. Found Comput Math 10, 303–338 (2010). https://doi.org/10.1007/s10208-009-9057-1

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  • DOI: https://doi.org/10.1007/s10208-009-9057-1

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