The Set of Orthogonal Tensor Trains

TL;DR

This paper investigates the set of tensors with orthogonal tensor train decompositions, revealing they are characterized by quadratic equations and an additional higher-degree equation, extending the class of orthogonally decomposable tensors.

Contribution

It introduces a new class of tensors extending orthogonally decomposable tensors and characterizes their defining equations, including quadratic and higher-degree equations.

Findings

Set of orthogonal tensor train tensors is defined by quadratic equations.

Extension of orthogonally decomposable tensors with additional higher-degree equation.

Provides algebraic characterization of these tensors.

Abstract

In this paper we study the set of tensors that admit a special type of decomposition called an orthogonal tensor train decomposition. Finding equations defining varieties of low-rank tensors is generally a hard problem, however, the set of orthogonally decomposable tensors is defined by appealing quadratic equations. The tensors we consider are an extension of orthogonally decomposable tensors. We show that they are defined by similar quadratic equations, as well as an interesting higher-degree additional equation.