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Deterministic Sparse Fourier Transform with an 𝓁_{∞} Guarantee

Authors Yi Li , Vasileios Nakos

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ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Streaming, sublinear and near linear time algorithms
Keywords
  • Fourier sparse recovery
  • derandomization
  • incoherent matrices

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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 𝓁₂/𝓁₁ guarantee. We focus on the stronger 𝓁_∞/𝓁₁ 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 𝓁_∞/𝓁₁ recovery in time O(nk log² n), and a deterministic collection of O(k² log² n) samples for the 𝓁_∞/𝓁₁ 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.

Cite As Get BibTex

Yi Li and Vasileios Nakos. Deterministic Sparse Fourier Transform with an 𝓁_{∞} Guarantee. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 77:1-77:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ICALP.2020.77

Author Details

Yi Li
  • Nanyang Technological University, Singapore, Singapore
Vasileios Nakos
  • Universität des Saarlandes, Saarbrücken, Germany
  • Max Planck Institut für Informatik, Saarland Informatics Campus, Saarbrücken, Germany

Funding

  • Nakos, Vasileios: This work is part of the project TIPEA that has received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No. 850979). Part of the work was completed when the author was a Ph.D. student in Harvard University and supported in part by NSF CAREER award CCF-1350670.

Acknowledgements

We would like to thank anonymous reviewers for their valuable feedback.

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