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arXiv · 1201.2501

Nearly Optimal Sparse Fourier Transform

Haitham Hassanieh·Piotr Indyk·Dina Katabi·Eric Price
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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^{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.

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Haitham Hassanieh, Piotr Indyk, Dina Katabi, Eric Price. 2012-04-06. Nearly Optimal Sparse Fourier Transform. https://arxiv.org/abs/1201.2501

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