A universal sampling method for reconstructing signals with simple Fourier transforms

Haim Avron, Michael Kapralov, Cameron Musco, Christopher Musco, Ameya Velingker, Amir Zandieh

Research output: Chapter in Book/Report/Conference proceedingConference contribution

5 Scopus citations

Abstract

Reconstructing continuous signals based on a small number of discrete samples is a fundamental problem across science and engineering. We are often interested in signals with “simple” Fourier structure – e.g., those involving frequencies within a bounded range, a small number of frequencies, or a few blocks of frequencies – i.e., bandlimited, sparse, and multiband signals, respectively. More broadly, any prior knowledge on a signal’s Fourier power spectrum can constrain its complexity. Intuitively, signals with more highly constrained Fourier structure require fewer samples to reconstruct. We formalize this intuition by showing that, roughly, a continuous signal from a given class can be approximately reconstructed using a number of samples proportional to the statistical dimension of the allowed power spectrum of that class. We prove that, in nearly all settings, this natural measure tightly characterizes the sample complexity of signal reconstruction. Surprisingly, we also show that, up to log factors, a universal nonuniform sampling strategy can achieve this optimal complexity for any class of signals. We present an efficient and general algorithm for recovering a signal from the samples taken. For bandlimited and sparse signals, our method matches the state-of-the-art, while providing the the first computationally and sample efficient solution to a broader range of problems, including multiband signal reconstruction and Gaussian process regression tasks in one dimension. Our work is based on a novel connection between randomized linear algebra and the problem of reconstructing signals with constrained Fourier structure. We extend tools based on statistical leverage score sampling and column-based matrix reconstruction to the approximation of continuous linear operators that arise in the signal reconstruction problem. We believe these extensions are of independent interest and serve as a foundation for tackling a broad range of continuous time problems using randomized methods.

Original languageEnglish (US)
Title of host publicationSTOC 2019 - Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing
EditorsMoses Charikar, Edith Cohen
PublisherAssociation for Computing Machinery
Pages1051-1063
Number of pages13
ISBN (Electronic)9781450367059
DOIs
StatePublished - Jun 23 2019
Event51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019 - Phoenix, United States
Duration: Jun 23 2019Jun 26 2019

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017

Conference

Conference51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019
CountryUnited States
CityPhoenix
Period6/23/196/26/19

All Science Journal Classification (ASJC) codes

  • Software

Keywords

  • Leverage score sampling
  • Numerical linear algebra
  • Signal reconstruction

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  • Cite this

    Avron, H., Kapralov, M., Musco, C., Musco, C., Velingker, A., & Zandieh, A. (2019). A universal sampling method for reconstructing signals with simple Fourier transforms. In M. Charikar, & E. Cohen (Eds.), STOC 2019 - Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing (pp. 1051-1063). (Proceedings of the Annual ACM Symposium on Theory of Computing). Association for Computing Machinery. https://doi.org/10.1145/3313276.3316363