Faster Algorithms for Sparse Fourier Transform

Transcription

1 Faster Algorithms for Sparse Fourier Transform Haitham Hassanieh Piotr Indyk Dina Katabi Eric Price MIT Material from: Hassanieh, Indyk, Katabi, Price, Simple and Practical Algorithms for Sparse Fourier Transform, SODA 12. Hassanieh, Indyk, Katabi, Price, Nearly Optimal Sparse Fourier Transform, STOC 12.

2 The Discrete Fourier Transform Discrete Fourier Transform: Given: a signal Goal: compute the frequency vector such that for 1 : Fundamental tool: / Compression (audio, image, video) Signal processing Data analysis Wireless Communication FFT : O log time Sampled Audio Data (Time) DFT of Audio Samples (Frequency)

3 Sparse Fourier Transform Time Domain Signal Sparse Frequency Spectrum Approximately Sparse Frequency Spectrum Often the Fourier transform is dominated by a small number of peaks Only few of the frequency coefficients are nonzero. An exactly k sparse signal has only k nonzero frequency coefficients. In practice : approximate a sparse signal using the k largest peaks. Problem : Can we recover the k sparse frequency spectrum faster than FFT?

4 Previous Work Algorithms: Boolean cube : [KM92], [GL89]. What about? Complex FT: [Mansour 92, GGIMS02, AGS03, GMS05, Iwen10, Aka10] Best running time: [GMS05] O In theory : Improves over FFT for / log 3 In Practice : Large constants; need / 40,000 to beat FFT Goal: Theory: improve over FFT for all values of Practice: faster runtime than FFT.

5 Our results Randomized algorithms, with constant probability of success Exactly sparse case, recover : log Optimal if FFT optimal Approximately sparse case, recover : Let Err min l 2 /l 2 guarantee Err : log log / Improves over FFT for any l /l 2 guarantee Err : log log Improves over FFT for /log

6 Sparse FFT Algorithm

7 Intuition n point DFT : log Time Domain Signal Frequency Domain n point DFT of first B terms : log Boxcar sinc Cut off Time signal Frequency Domain B point DFT of first B terms: log Alias Boxcar First B samples Frequency Domain Subsample sinc

8 Framework n point DFT of all n samples B point DFT of first B sample n frequencies hash into B buckets Hashes the Fourier coefficients into buckets in O( log ) time Issues Leakage : Subsample Filter sinc Given these buckets, how can we estimate the locations and values the large frequencies?

9 Filter: Sinc Boxcar Sinc Polynomial decay Leaking many buckets

10 Filter: Gaussian Gaussian Gaussian Exponential decay Leaking to buckets

11 Filters: Wider Gaussian Wider Gaussian Narrow Gaussian Exponential decay Leaking to <1 buckets But trivial contribution to the correct bucket

12 Filters: Sinc Gaussian Sinc Gaussian Boxcar Gaussian Boxcar size / : / frequencies hash into each bucket Still exponential decay Leaking to at most 1 bucket Sufficient contribution to the correct bucket /2, /2 Replace Gaussians with Dolph Chebyshev window functions

13 Finding the support = B point DFT = Subsample Assume no collisions: At most one large frequency hashes into each bucket. Large frequency hashes to bucket : = leakage Recall: DFT( / ) = = B point DFT : = / leakage

14 Finding the support = B point DFT = Subsample Assume no collisions: At most one large frequency hashes into each bucket. Large frequency hashes to bucket : = = / mod Find all frequencies in 2 log

15 Random Hashing Some Large frequencies collide: Subtract and recurs Small number of collisions converges in few iterations Every iteration needs new random hashing: Permute time domain signal permute frequency domain is invertible mod : = / = Permutation : mod

16 Iteration i : / 2 Algorithm Permute spectrum : = / = point DFT = Subsample Recover locations and values of large frequencies Each iteration takes O( iterations Total time : ) time.

17 Experiments (exactly k sparse algorithm)

18 Setup Similar to earlier work: Random 0 1 k sparse vectors Fix n, vary k Fix k, vary n

19 Experiments

20 Experiments, ctd

21 Conclusions Sparse FFT with running times : for exactly sparse case for approximately sparse case Improves over FFT for Significant improvement in practice time for approximately sparse signals? Not clear: log / samples needed, extra log for FT