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Learning Significant Fourier Coefficients over Finite Abelian Groups

2003; Akavia, Goldwasser, Safra

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Encyclopedia of Algorithms

Keywords and Synonyms

Learning heavy fourier coefficients; Finding heavy fourier coefficients                    

Problem Definition

Fourier transform is among the most widely used tools in computer science. Computing the Fourier transform of a signal of length N may be done in time \( { \Theta(N\log N) } \) using the Fast Fourier Transform (FFT) algorithm. This time bound clearly cannot be improved below \( { \Theta(N) } \), because the output itself is of length N. Nonetheless, it turns out that in many applications it suffices to find only the significant Fourier coefficients, i. e., Fourier coefficients occupying, say, at least \( { 1\% } \)of the energy of the signal. This motivates the problem discussed in this entry: the problem of efficiently finding and approximating the significant Fourier coefficients of a given signal (SFT, in short). A naive solution for SFT is to first compute the entire Fourier transform of the given signal and then to output only the significant Fourier coefficients;...

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Notes

  1. 1.

    For readers more accustomed to vector notation, the authors remark that there is a simple correspondence between vector and functional notation. For example, a one-dimensional signal \( { (v_1,\dots, v_N)\in \mathbb{C^N} } \) corresponds to the function \( { f\colon\mathbb{Z}_N\to\mathbb{C} } \) defined by \( { f(i)=v_i } \) for all \( { i=1,\dots, N } \). Likewise, a two-dimensional signal \( { M\in \mathbb{C}^{N_1\times N_2} } \) corresponds to the function \( { f\colon\mathbb{Z}_{N_1}\times\mathbb{Z}_{N_2}\to\mathbb{C} } \) defined by \( { f(i,j)=M_{ij} } \) for all \( { i=1,\dots, N_1 } \) and \( { j=1,\dots, N_2 } \).

  2. 2.

    Say that an algorithm is given oracle access to a function f over G, if it can request and receive the value f(x) for any \( { x\in G } \) in unit time.

  3. 3.

    \( { \mathcal{P}=\{{P_N\colon\mathbb{Z}_N\to{\{ 0,1 \}}}\}_{{N\in\mathbb{Z}^+}} } \) is a family of efficiently computable functions if there is an algorithm that given any \( { {N\in\mathbb{Z}^+} } \) and \( { x\in \mathbb{Z}_N } \) outputs \( { P_N(x) } \) in time \( { {poly}(\log N) } \).

  4. 4.

    A family of functions \( { \mathcal{P} =\{{P_N\colon \mathbb{Z}_N\to{\{ 0,1 \}}}\}_{N\in\mathbb{Z}^+} } \) is non-negligibly far from constant if \( { \forall {N\in\mathbb{Z}^+} } \) and \( { b\in{\{ 0,1 \}} } \), \( { \mathrm{Pr}_{j\in\mathbb{Z}_N}[P_N(j)=b]\le 1-{poly}(1/\log N) } \).

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Author information

Authors and Affiliations

  1. Department of Electrical Engineering and Computer Science, MIT, Cambridge, MA, USA

    Adi Akavia

Authors
  1. Adi Akavia
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Editor information

Editors and Affiliations

  1. Department of Electrical Engineering and Computer ScienceMcCormick School of Engineering and Applied Science, Northwestern University, Evanston, IL, 60208, USA

    Ming-Yang Kao Professor of Computer Science

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© 2008 Springer-Verlag

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Akavia, A. (2008). Learning Significant Fourier Coefficients over Finite Abelian Groups. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-30162-4_199

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