The Generalized Schur Decomposition and the rank-$R$ set of real $I\times J\times 2$ arrays

Abstract

It is known that a best low-rank approximation to multi-way arrays or higher-order tensors may not exist. This is due to the fact that the set of multi-way arrays with rank at most $R$ is not closed. Nonexistence of the best low-rank approximation results in diverging rank-1 components when an attempt is made to compute the approximation. Recently, a solution to this problem has been proposed for real $I\times J\times 2$ arrays. Instead of a best rank-$R$ approximation the best fitting Generalized Schur Decomposition (GSD) is computed. Under the restriction of nonsingular upper triangular matrices in the GSD, the set of GSD solutions equals the interior and boundary of the rank-$R$ set. Here, we show that this holds even without the restriction. We provide a complete classification of interior, boundary, and exterior points of the rank-$R$ set of real $I\times J\times 2$ arrays, and show that the set of GSD solutions equals the interior and boundary of this set.