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Improved Time Bounds for Near-Optimal Sparse Fourier Representations (2003)
| Venue: | in Proc. SPIE Wavelets XI |
| Citations: | 58 - 11 self |
BibTeX
@INPROCEEDINGS{Gilbert03improvedtime,
author = {A. C. Gilbert and S. Muthukrishnan and M. Strauss},
title = {Improved Time Bounds for Near-Optimal Sparse Fourier Representations},
booktitle = {in Proc. SPIE Wavelets XI},
year = {2003}
}
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Abstract
We study the problem of finding a Fourier representation R of B terms for a given discrete signal A of length N . The Fast Fourier Transform (FFT) can find the optimal N-term representation in O(N log N) time, but our goal is to get sublinear algorithms for B ! N , typically, B N . Suppose kAk2 M kRoptk 2 , where Ropt is the optimal output. The previously best known algorithms output R such that kA \Gamma Rk poly(B; log(1=ffi); log N; log M; 1=ffl): Even though this is sublinear in the input size, the dominating term is the polynomial factor in B which is B . In our experience, this is a limitation in practice. Our main result is a significantly improved algorithm for this problem. Our algorithms output R such that kA \Gamma Rk B \Delta poly(log(1=ffi); log N; log M; 1=ffl): We also obtain improvements for higher dimensional Fourier transforms. We need two crucial ideas to achieve this bound: bulk sampling and estimation for multipoint polynomial evaluation using an unevenly-spaced Fourier tranform, and construction and use of arithmeticprogression independent random variables. Our improved algorithms are likely to find many applications. 1
Keyphrases
near-optimal sparse fourier representation time bound algorithm output arithmeticprogression independent random variable many application crucial idea unevenly-spaced fourier tranform improved algorithm main result bulk sampling gamma rk poly fast fourier transform optimal output polynomial factor gamma rk delta poly sublinear algorithm dominating term discrete signal multipoint polynomial evaluation input size fourier representation optimal n-term representation dimensional fourier transforms
