Are diverging CP components always nearly proportional?

TL;DR

This paper investigates the behavior of diverging components in CP tensor decompositions, showing that in most cases, diverging components tend to become proportional, with some exceptions demonstrated through examples.

Contribution

It proves that diverging components in CP decompositions generally become proportional, providing partial proofs for larger groups and illustrating exceptional non-proportional cases.

Findings

Diverging components tend to become proportional almost everywhere.

Examples of non-proportional diverging components are rare and exceptional.

Partial proof provided for larger groups of diverging components.

Abstract

Fitting a Candecomp/Parafac (CP) decomposition (also known as Canonical Polyadic decomposition) to a multi-way array or higher-order tensor, is equivalent to finding a best low-rank approximation to the multi-way array or higher-order tensor, where the rank is defined as the outer-product rank. However, such a best low-rank approximation may not exist due to the fact that the set of multi-way arrays with rank at most $R$ is not closed for $R\ge 2$. Nonexistence of a best low-rank approximation results in (groups of) diverging rank-1 components when an attempt is made to compute the approximation. In this note, we show that in a group of two or three diverging components, the components converge to proportionality almost everywhere. A partial proof of this result for larger groups of diverging components is also given. Also, we give examples of groups of three, four, and six non-proportional diverging components. These examples are shown to be exceptional cases.