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A deterministic sparse FFT for functions with structured Fourier sparsity

  • Sina Bittens
  • Ruochuan Zhang
  • Mark A. Iwen
Article
  • 29 Downloads

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.

Keywords

Sparse Fourier transform (SFT) Structured sparsity Deterministic constructions Approximation algorithms 

Mathematics Subject Classification (2010)

05-04 42A10 42A15 42A16 42A32 65T40 65T50 68W25 94A12 

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Notes

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 information

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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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute for Numerical and Applied MathematicsUniversity of GöttingenGöttingenGermany
  2. 2.Department of MathematicsMichigan State UniversityEast LansingUSA
  3. 3.Department of Mathematics, and Department of Computational Mathematics, Science, and Engineering (CMSE)Michigan State UniversityEast LansingUSA

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