A New Class of Fully Discrete Sparse Fourier Transforms: Faster Stable Implementations with Guarantees

Merhi, Sami; Zhang, Ruochuan; Iwen, Mark A.; Christlieb, Andrew

doi:10.1007/s00041-018-9616-4

Cited by 31 publications

(35 citation statements)

References 29 publications

(96 reference statements)

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“…The presented algorithm requires sampling the signal at arbitrary locations in [0, 2π). A natural approach is to emulate sampling off-grid (i.e., at arbitrary points in [0, 2π)) given discrete samples that we have access to, which is achieved in [MZIC17] giving an O(k 2 ) time deterministic algorithm for one dimensional sparse FFT. But this is a challenging task in multi-dimensional setting for several reasons.…”

Section: Significance Of Our Results and Related Workmentioning

confidence: 99%

Dimension-independent Sparse Fourier Transform

Kapralov¹,

Velingker²,

Zandieh³

2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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The Discrete Fourier Transform (DFT) is a fundamental computational primitive, and the fastest known algorithm for computing the DFT is the FFT (Fast Fourier Transform) algorithm. One remarkable feature of FFT is the fact that its runtime depends only on the size N of the input vector, but not on the dimensionality of the input domain:The state of the art for Sparse FFT, i.e. the problem of computing the DFT of a signal that has at most k nonzeros in Fourier domain, is very different: all current techniques for sublinear time computation of Sparse FFT incur an exponential dependence on the dimension d in the runtime. In this paper we give the first algorithm that computes the DFT of a ksparse signal in time poly(k, log N ) in any dimension d, avoiding the curse of dimensionality inherent in all previously known techniques. Our main tool is a new class of filters that we refer to as adaptive aliasing filters: these filters allow isolating frequencies of a k-Fourier sparse signal using O(k) samples in time domain and O(k log N ) runtime per frequency, in any dimension d.We also investigate natural average case models of the input signal:(1) worst case support in Fourier domain with randomized coefficients and (2) random locations in Fourier domain with worst case coefficients. Our techniques lead to an O(k 2 ) time algorithm for the former and an O(k) time algorithm for the latter.

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Section: Significance Of Our Results and Related Workmentioning

confidence: 99%

Dimension-independent Sparse Fourier Transform

Kapralov¹,

Velingker²,

Zandieh³

2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…(round() means to make decimals rounded). The Equation (14) respecting the composition of filtered signal in bucket i can be obtained. The proof of this equation is same as the proof of Lemma 1 in paper [19] (In the figure, * represents the convolution of the signal and the filter in the frequency domain, × represents the multiplication of the signal and the filter in the frequency domain).…”

Section: The First Stage Of Sfft: Frequency Bucketizationmentioning

confidence: 99%

Empirical Evaluation of Typical Sparse Fast Fourier Transform Algorithms

Jiang

Chen

2020

Preprint

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Computing the Sparse Fast Fourier Transform(sFFT) of a K-sparse signal of size N has emerged as a critical topic for a long time. The sFFT algorithms decrease the runtime and sampling complexity by taking advantage of the signal's inherent characteristics that a large number of signals are sparse in the frequency domain. More than ten sFFT lgorithms have been proposed, which can be classfied into many types according to filter, framework, method of location, method of estimation. In this paper, the technology of these algorithms is completely analyzed in theory. The performance of them is thoroughly tested and verified in practice. The theoretical analysis includes the following contents: five operations of ignal, three methods of frequency bucketization, five methods of location, four methods of estimation, two problems caused by bucketization, three methods to solve these two problems, four algorithmic frameworks. All the above technologies and methods are introduced in detail and examples are given to illustrate the above research. After theoretical research, we make experiments for computing the signals of different SNR, N, K by a standard testing platform and record the run time, percentage of the signal sampled and L0;L1;L2 error with eight different sFFT algorithms. The result of experiments satisfies the inferences obtained in theory.

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“…In exchange for this slight unreliability in producing accurate output, the fastest of these randomized techniques are able to compute the Fourier series of n-sparse f in just n log O(1) N -time. The most efficient, numerically stable, and publicly available implementations of these methods are based on random algorithms developed out of MIT [17,19,23], Michigan [13,15,27], and Michigan State [10,32,45]. 1 We point the reader to a recent survey of such algorithms, techniques, and implementations for more details [14].…”

Section: Related Work: Sparse Fourier Transformsmentioning

confidence: 99%