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Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2020.77
URN: urn:nbn:de:0030-drops-124844
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12484/
Li, Yi ;
Nakos, Vasileios
Deterministic Sparse Fourier Transform with an ?_{∞} Guarantee
Abstract
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
@InProceedings{li_et_al:LIPIcs:2020:12484,
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 = {https://drops.dagstuhl.de/opus/volltexte/2020/12484},
URN = {urn:nbn:de:0030-drops-124844},
doi = {10.4230/LIPIcs.ICALP.2020.77},
annote = {Keywords: Fourier sparse recovery, derandomization, incoherent matrices}
}
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Keywords: |
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Fourier sparse recovery, derandomization, incoherent matrices |
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Collection: |
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47th International Colloquium on Automata, Languages, and Programming (ICALP 2020) |
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Issue Date: |
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2020 |
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Date of publication: |
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29.06.2020 |
