### 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 ^{2}R ^{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 ^{m}log ^{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 language | English |
---|---|

Pages (from-to) | 915-953 |

Number of pages | 39 |

Journal | Journal of Fourier Analysis and Applications |

Volume | 18 |

Issue number | 5 |

DOIs | |

Publication status | Published - 1 Oct 2012 |

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### Keywords

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

### Cite this

*Journal of Fourier Analysis and Applications*,

*18*(5), 915-953. https://doi.org/10.1007/s00041-012-9227-4

}

*Journal of Fourier Analysis and Applications*, vol. 18, no. 5, pp. 915-953. https://doi.org/10.1007/s00041-012-9227-4

**Superfast Fourier Transform Using QTT Approximation.** / Dolgov, Sergey; Khoromskij, Boris; Savostyanov, Dmitry.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Superfast Fourier Transform Using QTT Approximation

AU - Dolgov, Sergey

AU - Khoromskij, Boris

AU - Savostyanov, Dmitry

PY - 2012/10/1

Y1 - 2012/10/1

N2 - 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.

AB - 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.

KW - Convolution

KW - Fourier transform

KW - High-dimensional problems

KW - QTT

KW - Quantum Fourier transform

KW - Sparse Fourier transform

KW - Tensor train format

UR - http://www.scopus.com/inward/record.url?scp=84866519225&partnerID=8YFLogxK

U2 - 10.1007/s00041-012-9227-4

DO - 10.1007/s00041-012-9227-4

M3 - Article

AN - SCOPUS:84866519225

VL - 18

SP - 915

EP - 953

JO - Journal of Fourier Analysis and Applications

JF - Journal of Fourier Analysis and Applications

SN - 1069-5869

IS - 5

ER -