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A deterministic sparse FFT for functions with structured Fourier sparsity

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Abstract

In this paper, a deterministic sparse Fourier transform algorithm is presented which breaks the quadratic-in-sparsity runtime bottleneck for a large class of periodic functions exhibiting structured frequency support. These functions include, e.g., the often considered set of block frequency sparse functions of the form

$$f(x) = \sum\limits^{n}_{j = 1} \sum\limits^{B-1}_{k = 0} c_{\omega_{j} + k} e^{i(\omega_{j} + k)x},~~\{ \omega_{1}, \dots, \omega_{n} \} \subset \left( -\left\lceil \frac{N}{2}\right\rceil, \left\lfloor \frac{N}{2}\right\rfloor\right]\cap\mathbb{Z}$$

as a simple subclass. Theoretical error bounds in combination with numerical experiments demonstrate that the newly proposed algorithms are both fast and robust to noise. In particular, they outperform standard sparse Fourier transforms in the rapid recovery of block frequency sparse functions of the type above.

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Acknowledgements

The authors would like to thank both Felix Krahmer for introducing them at TUM in the summer of 2016 and Gerlind Plonka for her ongoing support, and particularly for her generosity in providing resources that aided in the writing of this paper.

Funding

Sina Bittens was supported in part by the DFG in the framework of the GRK 2088. Mark Iwen and Ruochuan Zhang were both supported in part by NSF DMS-1416752.

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Authors and Affiliations

  1. Institute for Numerical and Applied Mathematics, University of Göttingen, Lotzestr. 16-18, 37083, Göttingen, Germany

    Sina Bittens

  2. Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA

    Ruochuan Zhang

  3. Department of Mathematics, and Department of Computational Mathematics, Science, and Engineering (CMSE), Michigan State University, East Lansing, MI, 48824, USA

    Mark A. Iwen

Authors
  1. Sina Bittens
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  2. Ruochuan Zhang
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  3. Mark A. Iwen
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Correspondence to Sina Bittens.

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Communicated by: Yang Wang

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Bittens, S., Zhang, R. & Iwen, M.A. A deterministic sparse FFT for functions with structured Fourier sparsity. Adv Comput Math 45, 519–561 (2019). https://doi.org/10.1007/s10444-018-9626-4

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