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.

翻译：张量分解计算中的一个主要问题是如何选择秩一分量的数量，因为目前尚无有限算法可确定张量的秩。为此目的常用的一种方法是通过求解优化问题找到低维子空间，并假设分量数量固定。然而，尽管该算法高效且易于实现，但其常收敛至不良局部极小值，且易受异常值和噪声影响。本文旨在建立一个精确张量分解的数学框架，能够将张量表示为有限个低秩张量之和。本文将针对三个不同问题展开推导：i) 非负自伴张量算子的分解；ii) 线性张量变换的分解；iii) 一般张量的分解。