Superfast Fourier Transform Using QTT Approximation

Sergey Dolgov, Boris Khoromskij, Dmitry Savostyanov

Research output: Contribution to journalArticle

Abstract

We propose Fourier transform algorithms using QTT format for data-sparse approximate representation of one- and multi-dimensional vectors (m-tensors). Although the Fourier matrix itself does not have a low-rank QTT representation, it can be efficiently applied to a vector in the QTT format exploiting the multilevel structure of the Cooley-Tukey algorithm. The m-dimensional Fourier transform of an n×⋯×n vector with n=2 d has O(md 2R 3) complexity, where R is the maximum QTT-rank of input, output and all intermediate vectors in the procedure. For the vectors with moderate R and large n and m the proposed algorithm outperforms the O(n mlog n) fast Fourier transform (FFT) algorithm and has asymptotically the same log-squared complexity as the superfast quantum Fourier transform (QFT) algorithm. By numerical experiments we demonstrate the examples of problems for which the use of QTT format relaxes the grid size constrains and allows the high-resolution computations of Fourier images and convolutions in higher dimensions without the 'curse of dimensionality'. We compare the proposed method with Sparse Fourier transform algorithms and show that our approach is competitive for signals with small number of randomly distributed frequencies and signals with limited bandwidth.

Original languageEnglish
Pages (from-to)915-953
Number of pages39
JournalJournal of Fourier Analysis and Applications
Volume18
Issue number5
DOIs
Publication statusPublished - 1 Oct 2012

Keywords

  • Convolution
  • Fourier transform
  • High-dimensional problems
  • QTT
  • Quantum Fourier transform
  • Sparse Fourier transform
  • Tensor train format

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