Decomposition of linear tensor transformations
Claudio Turchetti
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.