Algebraic Wavelet Transform via Quantics Tensor Train Decomposition
Algebraic Wavelet Transform via Quantics Tensor Train Decomposition
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DOI：
10.1137/100811647
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发表时间：
2011-05
期刊：
影响因子：
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通讯作者：
I. Oseledets;E. Tyrtyshnikov
中科院分区：
文献类型：
--
作者：
I. Oseledets;E. Tyrtyshnikov
在本文中，我们表明，最近推出的量子张量序列（QTT）分解可以被认为是一个代数小波变换与自适应确定的过滤器。获得QTT分解的主要算法可以重新表述为寻找“好的子空间”或“好的基”的方法，并被认为是初始张量到稀疏张量的参数化变换。这种解释允许我们引入张量训练SVD（TT-SVD）算法的修改，使其在原始算法不工作的情况下工作;它导致新的小波变换，称为小波张量训练（WTT）变换。对WTT变换的性质进行了数值研究，提出了消失矩个数的理论猜想。结果表明，WTT变换是正交的建设，和效率的WTT相比，往往优于Daubechies小波变换的某些类的功能相关的向量和矩阵。
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.