Improved time bounds for near-optimal sparse Fourier representations

A. C. Gilbert, S. Muthukrishnan, M. Strauss

    Research output: Contribution to journalConference articlepeer-review

    Abstract

    We study the problem of finding a Fourier representation R of m terms for a given discrete signal A of length N. The Fast Fourier Transform (FFT) can find the optimal N-term representation in time O(N log N), but our goal is to get sublinear time algorithms when m " N. Suppose ∥A∥ 2 ≤ M ∥A - R opt2, where R opt is the optimal output. The previously best known algorithms output R such that ∥ A - R∥ 2 2 ≤ (1 + ε)∥ A - R opt2 2 with probability at least 1 - δ in time* polym,(m, log(1/δ), log N, log M, 1/ε). Although this is sublinear in the input size, the dominating expression is the polynomial factor in m which, for published algorithms, is greater than or equal to the bottleneck at m 2 that we identify below. Our experience with these algorithms shows that this is serious limitation in theory and in practice. Our algorithm beats this m 2 bottleneck. Our main result is a significantly improved algorithm for this problem and the d-dimensional analog. Our algorithm outputs an R with the same approximation guarantees but it runs in time m · poly(log(1/δ), log N, log M, 1/ε). A version of the algorithm holds for all N, though the details differ slightly according to the factorization of N. For the d-dimensional problem of size N 1 × N 2 × ⋯× N d, the linear-in-m algorithm extends efficiently to higher dimensions for certain factorizations of the N i's; we give a quadratic-in-m algorithm that works for any values of N i's. This article replaces several earlier, unpublished drafts.

    Original languageEnglish (US)
    Article number59141A
    Pages (from-to)1-15
    Number of pages15
    JournalProceedings of SPIE - The International Society for Optical Engineering
    Volume5914
    DOIs
    StatePublished - 2005
    EventWavelets XI - San Diego, CA, United States
    Duration: Jul 31 2005Aug 3 2005

    Keywords

    • Fourier analysis
    • Randomized approximation algorithms
    • Sampling
    • Sparse analysis

    ASJC Scopus subject areas

    • Electronic, Optical and Magnetic Materials
    • Condensed Matter Physics
    • Computer Science Applications
    • Applied Mathematics
    • Electrical and Electronic Engineering

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