Nearly optimal sparse fourier transform
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Katabi_Nearly Optimal.pdf
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298.54 KB
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Author(s)
Hassanieh, Haitham
Indyk, Piotr
Katabi, Dina
Price, Eric C.
Date Issued
May 2012
Journal
Proceedings of the 44th Symposium on Theory of Computing (STOC '12 )
Publisher
Association for Computing Machinery (ACM)
Citation
Hassanieh, Haitham et al. “Nearly Optimal Sparse Fourier Transform.” Proceedings of the 44th Symposium on Theory of Computing (STOC '12 ). 563.
Version
Author's final manuscript
Abstract
We consider the problem of computing the k-sparse approximation to the discrete Fourier transform of an n-dimensional signal. We show: An O(k log n)-time randomized algorithm for the case where the input signal has at most k non-zero Fourier coefficients, and An O(k log n log(n/k))-time randomized algorithm for general input signals.
Both algorithms achieve o(n log n) time, and thus improve over the Fast Fourier Transform, for any k=o(n). They are the first known algorithms that satisfy this property. Also, if one assumes that the Fast Fourier Transform is optimal, the algorithm for the exactly k-sparse case is optimal for any k = n[superscript Ω(1)].
We complement our algorithmic results by showing that any algorithm for computing the sparse Fourier transform of a general signal must use at least Ω(k log (n/k) / log log n) signal samples, even if it is allowed to perform adaptive sampling.
MIT Department
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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Creative Commons Attribution-Noncommercial-Share Alike 3.0
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DOI of Published Version
https://doi.org/10.1145/2213977.2214029
