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High-dimensional sparse Fourier algorithms

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Abstract

In this paper, we discuss the development of a sublinear sparse Fourier algorithm for high-dimensional data. In 11Adaptive Sublinear Time Fourier Algorithm” by Lawlor et al. (Adv. Adapt. Data Anal. 5(01):1350003, 2013), an efficient algorithm with \({\Theta }(k\log k)\) average-case runtime and Θ(k) average-case sampling complexity for the one-dimensional sparse FFT was developed for signals of bandwidth N, where k is the number of significant modes such that k ≪ N. In this work we develop an efficient algorithm for sparse FFT for higher dimensional signals, extending some of the ideas in Lawlor et al. (Adv. Adapt. Data Anal. 5(01):1350003, 2013). Note a higher dimensional signal can always be unwrapped into a one-dimensional signal, but when the dimension gets large, unwrapping a higher dimensional signal into a one-dimensional array is far too expensive to be realistic. Our approach here introduces two new concepts: “partial unwrapping” and “tilting.” These two ideas allow us to efficiently compute the sparse FFT of higher dimensional signals.

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Acknowledgments

We would like to thank Mark Iwen for his valuable advice.

Funding

This research is supported in part by AFOSR grants FA9550-11-1-0281, FA9550-12-1-0343 and FA9550-12-1-0455, NSF grant DMS-1115709, and MSU Foundation grant SPG-RG100059, as well as Hong Kong Research Grant Council grants 16306415, 16308518 and 16317416.

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Authors and Affiliations

  1. The Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX, USA

    Bosu Choi

  2. Computational Mathematics Science and Engineering, Michigan State University, East Lansing, MI, USA

    Andrew Christlieb

  3. Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

    Yang Wang

Authors
  1. Bosu Choi
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  2. Andrew Christlieb
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  3. Yang Wang
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Correspondence to Bosu Choi.

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Choi, B., Christlieb, A. & Wang, Y. High-dimensional sparse Fourier algorithms. Numer Algor 87, 161–186 (2021). https://doi.org/10.1007/s11075-020-00962-1

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