Are diverging CP components always nearly proportional?

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.

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