Theoretical and experimental analysis of a randomized algorithm for Sparse Fourier transform analysis

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• 作者：Jing Zou ; Anna Gilbert ; Martin Strauss and Ingrid Daubechies
• 关键词：RAℓ ; SFA ; Sparse Fourier representation ; Fast Fourier transform ; Sublinear algorithm ; Randomized algorithm
• 刊名：Journal of Computational Physics
• 出版年：2006
• 出版时间：2006
• 年：2006
• 卷：211
• 期：2
• 页码：572-595
• 全文大小：330 K

文摘

We analyze a sublinear RAℓSFA (randomized algorithm for Sparse Fourier analysis) that finds a near-optimal B-term Sparse representation R for a given discrete signal S of length N, in time and space , following the approach given in [A.C. Gilbert, S. Guha, P. Indyk, S. Muthukrishnan, M. Strauss, Near-Optimal Sparse Fourier Representations via Sampling, STOC, 2002]. Its time cost poly(log(N)) should be compared with the superlinear time requirement of the Fast Fourier Transform (FFT). A straightforward implementation of the RAℓSFA, as presented in the theoretical paper [A.C. Gilbert, S. Guha, P. Indyk, S. Muthukrishnan, M. Strauss, Near-Optimal Sparse Fourier Representations via Sampling, STOC, 2002], turns out to be very slow in practice. Our main result is a greatly improved and practical RAℓSFA. We introduce several new ideas and techniques that speed up the algorithm. Both rigorous and heuristic arguments for parameter choices are presented. Our RAℓSFA constructs, with probability at least 1-δ, a near-optimal B-term representation R in time poly(B)log(N)log(1/δ)/²log(M) such that . Furthermore, this RAℓSFA implementation already beats the FFTW for not unreasonably large N. We extend the algorithm to higher dimensional cases both theoretically and numerically. The crossover point lies at N70,000 in one dimension, and at N900 for data on a N×N grid in two dimensions for small B signals where there is noise.