Oseledets I.V., Tyrtyshnikov E.E. Algebraic wavelet transform via quantics tensor train decomposition, SIAM Journal of Scientific Computing, 33 (3), .1315-1328, 2011.

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

: tensor train format, quantics tensor train format, discrete wavelet transform