Decomposition of linear tensor transformations

TL;DR

This paper develops a mathematical framework for exact tensor decomposition, addressing the challenge of determining tensor rank and proposing methods for decomposing various types of tensors, including non-negative self-adjoint and linear tensor transformations.

Contribution

It introduces a novel framework for exact tensor decomposition applicable to different tensor classes, overcoming limitations of existing approximation methods.

Findings

Framework enables exact decomposition of certain tensor classes.

Methods handle non-negative self-adjoint tensors and linear transformations.

Addresses issues with traditional rank determination and local minima.

Abstract

One of the main issues in computing a tensor decomposition is how to choose the number of rank-one components, since there is no finite algorithms for determining the rank of a tensor. A commonly used approach for this purpose is to find a low-dimensional subspace by solving an optimization problem and assuming the number of components is fixed. However, even though this algorithm is efficient and easy to implement, it often converges to poor local minima and suffers from outliers and noise. The aim of this paper is to develop a mathematical framework for exact tensor decomposition that is able to represent a tensor as the sum of a finite number of low-rank tensors. In the paper three different problems will be carried out to derive: i) the decomposition of a non-negative self-adjoint tensor operator; ii) the decomposition of a linear tensor transformation; iii) the decomposition of a generic tensor.