Algebraic wavelet transform via quantics tensor train decomposition

Ivan V. Oseledets, Eugene E. Tyrtyshnikov

Research output: Contribution to journalArticlepeer-review

14 Citations (Scopus)


In this paper we show that recently introduced quantics tensor train (QTT) decomposition can be considered as an algebraic wavelet transform with adaptively determined filters. The main algorithm for obtaining QTT decomposition can be reformulated as a method seeking "good subspaces" or "good bases" and considered as a parameterized transformation of an initial tensor into a sparse tensor. This interpretation allows us to introduce a modification of the tensor train-SVD (TT-SVD) algorithm to make it work in cases where the original algorithm does not work; it results in the new wavelet-like transforms called wavelet tensor train (WTT) transform. Properties of WTT transforms are studied numerically, and a theoretical conjecture on the number of vanishing moments is proposed. It is shown that WTT transforms are orthogonal by construction, and the efficiency of WTT is compared with and often outperforms Daubechies wavelet transforms on certain classes of function-related vectors and matrices.

Original languageEnglish
Pages (from-to)1315-1328
Number of pages14
JournalSIAM Journal on Scientific Computing
Issue number3
Publication statusPublished - 2011
Externally publishedYes


  • Discrete wavelet transform
  • Quantics tensor train format
  • Tensor train format


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