## Simple and Practical Algorithm for Sparse Fourier Transform

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Citations: | 13 - 5 self |

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@MISC{Hassanieh_simpleand,

author = {Haitham Hassanieh and Piotr Indyk and Dina Katabi and Eric Price},

title = {Simple and Practical Algorithm for Sparse Fourier Transform},

year = {}

}

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### Abstract

We consider the sparse Fourier transform problem: given a complex vector x of length n, and a parameter k, estimate the k largest (in magnitude) coefficients of the Fourier transform of x. The problem is of key interest in several areas, including signal processing, audio/image/video compression, and learning theory. We propose a new algorithm for this problem. The algorithm leverages techniques from digital signal processing, notably Gaussian and Dolph-Chebyshev filters. Unlike the typical approach to this problem, our algorithm is not iterative. That is, instead of estimating “large ” coefficients, subtracting them and recursing on the reminder, it identifies and estimates the k largest coefficients in “one shot”, in a manner akin to sketching/streaming algorithms. The resulting algorithm is structurally simpler than its predecessors. As a consequence, we are able to extend considerably the range of sparsity, k, for which the algorithm is faster than FFT, both in theory and practice. 1

### Citations

1716 | Compressed sensing - Donoho - 2005 |

1298 | Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information - Candes, Romberg, et al. - 2006 |

415 | Efficient similarity search in sequence databases
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Citation Context ...lysis include computational learning theory [KM91, LMN93], analysis of boolean functions [KKL88, O’D08], multiscale analysis [DRZ07], compressed sensing [Don06, CRT06], similarity search in databases =-=[AFS93]-=-, spectrum sensing for wideband channels [LVS11], and datacenter monitoring [MNL10]. When the output of the DFT is sparse or approximately sparse, one can hope for an “output-sensitive” algorithm, who... |

278 | Constant depth circuits, Fourier transform, and learnability - Linial, Mansour, et al. - 1993 |

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184 | Learning decision trees using the Fourier spectrum
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Citation Context ...orithm can be expressed succinctly, and one can hope for an algorithm whose runtime is sublinear in the signal size n. The first such sublinear algorithm (for the Hadamard transform) was presented in =-=[KM91]-=-. Shortly after, several sublinear algorithms for the Fourier transform over the complex field were discovered [Man92, GGI + 02, AGS03, GMS05, Iwe10a, Aka10]. 2 These algorithms have a runtime that is... |

78 | Near-optimal sparse fourier representations via sampling - Gilbert, Guha, et al. - 2002 |

77 |
Data driven signal processing: An approach for energy efficient computing
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Citation Context ...o, i.e., the output of the DFT is (approximately) sparse. For example, a typical 8x8 block in a video frame has on average 7 non-negligible coefficients (i.e., 89% of the coefficients are negligible) =-=[CGX96]-=-. Images and audio data are equally sparse. This sparsity provides the rationale underlying compression schemes such as MPEG and JPEG. Other applications of sparse Fourier analysis include computation... |

65 | Combinatorial algorithms for compressed sensing
- Cormode, Muthukrishnan
- 2006
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Citation Context ...n” with period p. This filter does not leak: it is equal to 1 on 1/p fraction of coordinates and is zero elsewhere. Unfortunately, however, such a filter requires that p di4 This fact was implicit in =-=[CM06]-=-. For an explicit statement and proof see [GI10], remarks after Theorem 2. 5 One can randomize the positions of the frequencies by sampling the signal in time domain appropriately [GGI + 02, GMS05]. S... |

40 | Randomized interpolation and approximation of sparse polynomials
- Mansour
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(Show Context)
Citation Context ...et kernel, whose tail decays in an inverse linear fashion. Since the tail decays slowly, the Fourier coefficients binned to a particular bucket “leak” into other buckets. On the other hand, the paper =-=[Man92]-=- estimates the convolution in time domain via random sampling, which also leads to a large estimation error. To reduce these errors and obtain the ℓ2/ℓ2 guarantee, these algorithms have to perform mul... |

30 | Some topics in analysis of Boolean functions - O’Donnell - 2008 |

28 | Sparse recovery using sparse matrices
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Citation Context ...is equal to 1 on 1/p fraction of coordinates and is zero elsewhere. Unfortunately, however, such a filter requires that p di4 This fact was implicit in [CM06]. For an explicit statement and proof see =-=[GI10]-=-, remarks after Theorem 2. 5 One can randomize the positions of the frequencies by sampling the signal in time domain appropriately [GGI + 02, GMS05]. See section 3 part (b) for the description.vides... |

26 | Proving Hard-Core Predicates using List Decoding - Akavia, Goldwasser, et al. - 1995 |

17 |
Discrete-time signal processing. Upper Saddle River
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Citation Context ... supp(x) refers to the support of vector x, i.e., the set of non-zero coordinates. In this paper we assume that n is an integer power of 2. 3 Basics (a) Window Functions. In digital signal processing =-=[OSB99]-=- one defines window functions in the following manner: Definition 3.1. We define a (ǫ,δ,w) standard window function to be a symmetric vector F ∈ R n with supp(F) ⊆ [−w/2,w/2] such that ˆ F0 = 1, ˆ Fi ... |

15 | Combinatorial sublinear-time Fourier algorithms
- Iwen
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Citation Context ...mortizes this process but it requires solving instances of a non-uniform Fourier transform, which is expensive in practice. Interpolation-based algorithms are less common and limited to the design in =-=[Iwe10a]-=-. This approach uses a leakage-free filter, G, to avoid the need for iteration. Specifically, the algorithm in [Iwe10a] uses for G a filter in which Gi = 1 iff i mod n/p = 0 and Gi = 0 otherwise. The ... |

14 | Fast approximate correlation for massive time-series data
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Citation Context ...tions [KKL88, O’D08], multiscale analysis [DRZ07], compressed sensing [Don06, CRT06], similarity search in databases [AFS93], spectrum sensing for wideband channels [LVS11], and datacenter monitoring =-=[MNL10]-=-. When the output of the DFT is sparse or approximately sparse, one can hope for an “output-sensitive” algorithm, whose runtime depends on k, the number of computed “large” coefficients. Formally, giv... |

6 |
A sparse spectral method for homogenization multiscale problems. Multiscale Model
- Daubechies, Runborg, et al.
(Show Context)
Citation Context ...ion schemes such as MPEG and JPEG. Other applications of sparse Fourier analysis include computational learning theory [KM91, LMN93], analysis of boolean functions [KKL88, O’D08], multiscale analysis =-=[DRZ07]-=-, compressed sensing [Don06, CRT06], similarity search in databases [AFS93], spectrum sensing for wideband channels [LVS11], and datacenter monitoring [MNL10]. When the output of the DFT is sparse or ... |

6 |
Improved time bounds for near-optimal space Fourier representations
- Gilbert, Muthukrishnan, et al.
- 2005
(Show Context)
Citation Context ...The exponents of the polynomials, however, are typically large. The fastest among these algorithms have a runtime of the form O(k 2 log c n) (as in [GGI + 02, Iwe10a]) or the form O(k log c n))(as in =-=[GMS05]-=-), for some constant c > 2. In practice, the exponents in the runtime of these algorithms and their complex structures limit their ap1The algorithm in this paper has a somewhat stronger guarantee; see... |

5 |
Empirical evaluation of a sub-linear time sparse dft algorithm
- Iwen, Gilbert, et al.
(Show Context)
Citation Context ...rically against FFTW, an efficient implementation of the FFT with a runtime O(nlog n) [Iwe10a, IGS07]. The results show that the algorithm in [GMS05] is competitive with FFTW for n = 2 22 and k ≤ 135 =-=[IGS07]-=-. The algorithms in [GGI + 02, Iwe10a] require an even sparser signal (i.e., larger n and smaller k) to be competitive with FFTW. Results. In this paper, we propose a new sublinear algorithm for spars... |

5 | Improved approximation guarantees for sublinear-time Fourier algorithms
- Iwen
- 2012
(Show Context)
Citation Context ...2 2/k + δ‖x‖ 2 1 with probability 1 − 1/n. The additive term that depends on δ appears in all past algorithms [Man92, GGI + 02, AGS03, GMS05, Iwe10a, Aka10], although typically (with the exception of =-=[Iwe10b]-=-) it is eliminated by assuming that all coordinates are integers in the range {−n O(1) ...n O(1) } . In this paper, we keep the dependence on δ explicit. The ℓ∞/ℓ2 guarantee of Equation (1.2) is stron... |

3 | Sublinear time, measurement-optimal, sparse recovery for all
- Porat, Strauss
- 2012
(Show Context)
Citation Context ...cient. Further, once a large coefficient is isolated in a bucket, one needs to identify its frequency. In contrast to past work which typically uses binary search for this task, we adopt an idea from =-=[PS10]-=- and tailor it to our problem. Specifically, we simply select the set of “large” bins which are likely to contain large coefficients, and directly estimate all frequencies in those bins. To balance th... |

2 | Proving hard-core predicates using list decoding, FOCS ’03 - Akavia, Goldwasser, et al. - 2003 |

2 | A tutorial on fast Fourier sampling. Signal Processing Magazine - Gilbert, Strauss, et al. - 2008 |

2 |
Chong Meng Samson See. A new flexible filter bank for low complexity spectrum sensing in cognitive radios
- Lin, Vinod
- 2011
(Show Context)
Citation Context ...1, LMN93], analysis of boolean functions [KKL88, O’D08], multiscale analysis [DRZ07], compressed sensing [Don06, CRT06], similarity search in databases [AFS93], spectrum sensing for wideband channels =-=[LVS11]-=-, and datacenter monitoring [MNL10]. When the output of the DFT is sparse or approximately sparse, one can hope for an “output-sensitive” algorithm, whose runtime depends on k, the number of computed ... |

2 | Deterministic sparse fourier approximation via fooling arithmetic progressions - Akavia - 2010 |

1 |
AAFFT (Ann Arbor Fast Fourier Transform). http://sourceforge.net/projects/aafftannarborfa
- Iwen
- 2008
(Show Context)
Citation Context ...f sFFT 1.0 and sFFT 2.0, and compare them against two baselines: 1) FFTW 3.2.2 [FJ], which is the fastest public implementation for computing the DFT and has a runtime of O(nlog(n)), and 2) AAFFT 0.9 =-=[Iwe08]-=-, which is the prior sublinear algorithm with the fastest empirical runtime [IGS07]. For completeness, we compare against two variants of FFTW: basic and optimized. The optimized version requires prep... |