Dimension-independent Sparse Fourier Transform

Kapralov, Michael; Velingker, Ameya; Zandieh, Amir

doi:10.1137/1.9781611975482.168

Cited by 19 publications

(40 citation statements)

References 24 publications

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“…In most buckets a = 0, in a part of buckets a = 1, only in a small part of buckets a >= 2. Then the formula(21) can be translated to the formula (25), the problem to reconstruct spectrum is translated to how to calculate 2a variables as follows:…”

Section: E the Mpsft Algorithm By The Iterative Frameworkmentioning

confidence: 99%

“…The paper [17] proposes an overview of sFFT technology and summarizes a three-step approach in the stage of spectrum reconstruction and provides a standard testing platform that can be used to evaluate different sFFT algorithms. There are also some researches try to conquer the sFFT problem from a lot of aspects: computational complexity [18], [19], performance of the algorithm [20], [21], software [22], [23], higher dimensions [24], [25], implementation [26], hardware [27] and special setting [28], [29] perspectives.…”

Section: Introductionmentioning

confidence: 99%

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On Performance of Sparse Fast Fourier Transform Algorithms Using the Flat Window Filter

Jiang

Chen

2020

IEEE Access

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The problem of computing the Sparse Fast Fourier Transform(sFFT) of a K-sparse signal of size N has received significant attention for a long time. The first stage of sFFT is hashing the frequency coefficients into B(≈ K) buckets named frequency bucketization. The process of frequency bucketization is achieved through the use of filters: Dirichlet kernel filter, aliasing filter, flat filter, etc. The frequency bucketization through these filters can decrease runtime and sampling complexity in low dimensions. It is a hot topic about sFFT algorithms using the flat filter because of its convenience and efficiency since its emergence and wide application. The next stage of sFFT is the spectrum reconstruction by identifying frequencies that are isolated in their buckets. Up to now, there are more than thirty different sFFT algorithms using the sFFT idea as mentioned above by their unique methods. An important question now is how to analyze and evaluate the performance of these sFFT algorithms in theory and practice. In this paper, it is mainly discussed about sFFT algorithms using the flat filter. In the first part, the paper introduces the techniques in detail, including two types of frameworks, five different methods to reconstruct spectrum and corresponding algorithms. We get the conclusion of the performance of these five algorithms, including runtime complexity, sampling complexity and robustness in theory. In the second part, we make three categories of experiments for computing the signals of different SNR, different N , and different K by a standard testing platform and record the run time, percentage of the signal sampled, and L 0 , L 1 , L 2 error both in the exactly sparse case and general sparse case. The result of experiments is consistent with the inferences obtained in theory. It can help us to optimize these algorithms and use them correctly in the right areas.

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Section: E the Mpsft Algorithm By The Iterative Frameworkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On Performance of Sparse Fast Fourier Transform Algorithms Using the Flat Window Filter

Jiang

Chen

2020

IEEE Access

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“…The paper [21] summarizes a three-step approach in the stage of spectrum reconstruction and provides a standard testing platform to evaluate different sFFT algorithms. There have also been some researches that tried to conquer the sFFT problem from other aspects: complexity [22,23], performance [24,25], software [26,27], hardware [28], higher dimensions [29,30], implementation [31,32], and special setting [33,34] perspectives.…”

Section: Introductionmentioning

confidence: 99%

On Performance of Sparse Fast Fourier Transform Algorithms Using the Aliasing Filter

Jiang

Chen

2021

Electronics

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Computing the sparse fast Fourier transform (sFFT) has emerged as a critical topic for a long time because of its high efficiency and wide practicability. More than twenty different sFFT algorithms compute discrete Fourier transform (DFT) by their unique methods so far. In order to use them properly, the urgent topic of great concern is how to analyze and evaluate the performance of these algorithms in theory and practice. This paper mainly discusses the technology and performance of sFFT algorithms using the aliasing filter. In the first part, the paper introduces the three frameworks: the one-shot framework based on the compressed sensing (CS) solver, the peeling framework based on the bipartite graph and the iterative framework based on the binary tree search. Then, we obtain the conclusion of the performance of six corresponding algorithms: the sFFT-DT1.0, sFFT-DT2.0, sFFT-DT3.0, FFAST, R-FFAST, and DSFFT algorithms in theory. In the second part, we make two categories of experiments for computing the signals of different SNRs, different lengths, and different sparsities by a standard testing platform and record the run time, the percentage of the signal sampled, and the L0, L1, and L2 errors both in the exactly sparse case and the general sparse case. The results of these performance analyses are our guide to optimize these algorithms and use them selectively.

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“…However, it again suffers from the curse of dimensionality if it is extended to the higher-dimensional setting. In [17] this problem is resolved so that the multi-dimensional SFT with O(s 3 d 2 log 2 s log 2 N ) runtime and sampling complexity for any exactly s-sparse signal is developed. In [21] a general d-dimensional sparse Fourier algorithm was developed to achieve O(ds 2 N ) samples and O(ds 3 + ds 2 N log(sN )) runtime complexity.…”

Section: Introductionmentioning

confidence: 99%

Multiscale high-dimensional sparse Fourier algorithms for noisy data

Choi

Christlieb

Wang

2021

Mathematics, Computation and Geometry of Data

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We develop an efficient and robust high-dimensional sparse Fourier algorithm for noisy samples. Earlier in the paper High-dimensional sublinear sparse Fourier algorithms (2016) [3], an efficient sparse Fourier algorithm with Θ(ds log s) average-case runtime and Θ(ds) sampling complexity under certain assumptions was developed for signals that are s-sparse and bandlimited in the d-dimensional Fourier domain, i.e. there are at most s energetic frequencies and they are in [−N/2, N/2) d ∩ Z d . However, in practice the measurements of signals often contain noise, and in some cases may only be nearly sparse in the sense that they are well approximated by the best s Fourier modes. In this paper, we propose a multiscale sparse Fourier algorithm for noisy samples that proves to be both robust against noise and efficient.

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Dimension-independent Sparse Fourier Transform

Cited by 19 publications

References 24 publications

On Performance of Sparse Fast Fourier Transform Algorithms Using the Flat Window Filter

On Performance of Sparse Fast Fourier Transform Algorithms Using the Flat Window Filter

On Performance of Sparse Fast Fourier Transform Algorithms Using the Aliasing Filter

Multiscale high-dimensional sparse Fourier algorithms for noisy data

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