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DOI: 10.4230/LIPIcs.ICALP.2020.77
URN: urn:nbn:de:0030-drops-124844
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Li, Yi ; Nakos, Vasileios

Deterministic Sparse Fourier Transform with an ?_{∞} Guarantee

LIPIcs-ICALP-2020-77.pdf (0.5 MB)


In this paper we revisit the deterministic version of the Sparse Fourier Transform problem, which asks to read only a few entries of x ∈ ℂⁿ and design a recovery algorithm such that the output of the algorithm approximates x̂, the Discrete Fourier Transform (DFT) of x. The randomized case has been well-understood, while the main work in the deterministic case is that of Merhi et al. (J Fourier Anal Appl 2018), which obtains O(k² log^(-1) k ⋅ log^5.5 n) samples and a similar runtime with the ?₂/?₁ guarantee. We focus on the stronger ?_∞/?₁ guarantee and the closely related problem of incoherent matrices. We list our contributions as follows.
1) We find a deterministic collection of O(k² log n) samples for the ?_∞/?₁ recovery in time O(nk log² n), and a deterministic collection of O(k² log² n) samples for the ?_∞/?₁ sparse recovery in time O(k² log³n).
2) We give new deterministic constructions of incoherent matrices that are row-sampled submatrices of the DFT matrix, via a derandomization of Bernstein’s inequality and bounds on exponential sums considered in analytic number theory. Our first construction matches a previous randomized construction of Nelson, Nguyen and Woodruff (RANDOM'12), where there was no constraint on the form of the incoherent matrix.
Our algorithms are nearly sample-optimal, since a lower bound of Ω(k² + k log n) is known, even for the case where the sensing matrix can be arbitrarily designed. A similar lower bound of Ω(k² log n/ log k) is known for incoherent matrices.

BibTeX - Entry

  author =	{Yi Li and Vasileios Nakos},
  title =	{{Deterministic Sparse Fourier Transform with an ?_{∞} Guarantee}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{77:1--77:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-138-2},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{168},
  editor =	{Artur Czumaj and Anuj Dawar and Emanuela Merelli},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-124844},
  doi =		{10.4230/LIPIcs.ICALP.2020.77},
  annote =	{Keywords: Fourier sparse recovery, derandomization, incoherent matrices}

Keywords: Fourier sparse recovery, derandomization, incoherent matrices
Collection: 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)
Issue Date: 2020
Date of publication: 29.06.2020

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