Quartic Samples Suffice for Fourier Interpolation

Zhao Song, Baocheng Sun, Omri Weinstein, Ruizhe Zhang

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


We study the problem of interpolating a noisy Fourier-sparse signal in the time duration [0, T] from noisy samples in the same range, where the ground truth signal can be any k-Fourier-sparse signal with band-limit [-F, F]. Our main result is an efficient Fourier Interpolation algorithm that improves the previous best algorithm by [Chen, Kane, Price, and Song, FOCS 2016] in the following three aspects:•The sample complexity is improved from õ(k51) to õ(k4).•The time complexity is improved from õ(k10 Ω+40) to õ(k4 Ω).•The output sparsity is improved from õ(k10) to õ(k4). Here, Ω denotes the exponent of fast matrix multiplication. The state-of-the-art sample complexity of this problem is ∼ k4, but was only known to be achieved by an exponential-time algorithm. Our algorithm uses the same number of samples but has a polynomial runtime, laying the groundwork for an efficient Fourier Interpolation algorithm.The centerpiece of our algorithm is a new spectral analysis tool-the Signal Equivalent Method-which utilizes the structure of Fourier signals to establish nearly-optimal energy properties, and is the key for efficient and accurate frequency estimation. We use this method, along with a new sufficient condition for frequency recovery (a new high SNR band condition), to design a cheap algorithm for estimating 'significant' frequencies within a narrow range. Together with a signal estimation algorithm, we obtain a new Fourier Interpolation algorithm for reconstructing the ground-truth signal.

Original languageAmerican English
Title of host publicationProceedings - 2023 IEEE 64th Annual Symposium on Foundations of Computer Science, FOCS 2023
PublisherIEEE Computer Society
Number of pages12
ISBN (Electronic)9798350318944
StatePublished - 2023
Event64th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2023 - Santa Cruz, United States
Duration: 6 Nov 20239 Nov 2023

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS


Conference64th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2023
Country/TerritoryUnited States
CitySanta Cruz

All Science Journal Classification (ASJC) codes

  • General Computer Science


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