Theoretical and experimental analysis of a randomized algorithm for sparse Fourier transform analysis

  • Zou J
  • Gilbert A
  • Strauss M
 et al. 
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Abstract

We analyze a sublinear RAlSFA (Randomized Algorithm for Sparse Fourier Analysis) that finds a near-optimal B-term Sparse Representation R for a given discrete signal S of length N, in time and space poly(B,log(N)), following the approach given in \cite{GGIMS}. Its time cost poly(log(N)) should be compared with the superlinear O(N log N) time requirement of the Fast Fourier Transform (FFT). A straightforward implementation of the RAlSFA, as presented in the theoretical paper \cite{GGIMS}, turns out to be very slow in practice. Our main result is a greatly improved and practical RAlSFA. We introduce several new ideas and techniques that speed up the algorithm. Both rigorous and heuristic arguments for parameter choices are presented. Our RAlSFA constructs, with probability at least 1-delta, a near-optimal B-term representation R in time poly(B)log(N)log(1/delta)/ epsilon^{2} log(M) such that ||S-R||^{2}

Author-supplied keywords

  • fast fourier transform
  • random-
  • raℓsfa
  • sparse fourier representation
  • sublinear algorithm

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Authors

  • Jing Zou

  • Anna Gilbert

  • Martin Strauss

  • Ingrid Daubechies

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