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High-Dimensional Sparse Fourier Algorithms (1606.07407v4)

Published 23 Jun 2016 in math.NA and cs.NA

Abstract: In this paper, we discuss the development of a sublinear sparse Fourier algorithm for high-dimensional data. In Adaptive Sublinear Time Fourier Algorithm" by D. Lawlor, Y. Wang and A. Christlieb (2013), an efficient algorithm with $\Theta(k\log k)$ average-case runtime and $\Theta(k)$ average-case sampling complexity for the one-dimensional sparse FFT was developed for signals of bandwidth $N$, where $k$ is the number of significant modes such that $k\ll N$. In this work we develop an efficient algorithm for sparse FFT for higher dimensional signals, extending some of the ideas in the paper mentioned above. Note a higher dimensional signal can always be unwrapped into a one dimensional signal, but when the dimension gets large, unwrapping a higher dimensional signal into a one dimensional array is far too expensive to be realistic. Our approach here introduces two new concepts:partial unwrapping'' and ``tilting''. These two ideas allow us to efficiently compute the sparse FFT of higher dimensional signals.

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