2016/06/23 by Choi, Bosu, Christlieb, Andrew, Wang, Yang
#FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.1606.07407
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 Θ(klog k) average-case runtime and Θ(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≪ 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.