Orthogonal Decomposition of Tensor Trains

TL;DR

This paper investigates orthogonal tensor train decompositions, proposing algorithms for symmetric and non-symmetric cases, including whitening procedures and matrix decomposition methods, advancing tensor network analysis.

Contribution

It introduces new algorithms for orthogonal tensor train decomposition, including whitening techniques and matrix factorization approaches, addressing symmetric and non-symmetric cases.

Findings

Decomposition via random linear combinations for symmetric tensors.

Whitening transforms non-orthogonal to orthogonal tensor train problems.

Reduction to matrix decomposition for non-symmetric tensor trains.

Abstract

In this paper we study the problem of decomposing a given tensor into a tensor train such that the tensors at the vertices are orthogonally decomposable. When the tensor train has length two, and the orthogonally decomposable tensors at the two vertices are symmetric, we recover the decomposition by considering random linear combinations of slices. Furthermore, if the tensors at the vertices are symmetric and low-rank but not orthogonally decomposable, we show that a whitening procedure can transform the problem into the orthogonal case. When the tensor network has length three or more and the tensors at the vertices are symmetric and orthogonally decomposable, we provide an algorithm for recovering them subject to some rank conditions. Finally, in the case of tensor trains of length two in which the tensors at the vertices are orthogonally decomposable but not necessarily symmetric, we show that the decomposition problem reduces to the novel problem of decomposing a matrix into an orthogonal matrix multiplied by diagonal matrices on either side. We provide and compare two solutions, one based on Sinkhorn's theorem and one on Procrustes' algorithm. We conclude with a multitude of open problems in linear and multilinear algebra that arose in our study.