On best rank-2 and rank-(2,2,2) approximations of order-3 tensors

TL;DR

This paper investigates the existence of best rank-2 approximations for order-3 tensors, showing that such an approximation exists if and only if a certain local minimum condition is met, with implications for tensor approximation theory.

Contribution

It establishes a criterion linking the existence of best rank-2 approximations to local minima of the rank-(2,2,2) problem for real tensors, providing a practical check.

Findings

Best rank-2 approximation exists iff all rank-(2,2,2) minima have rank 2.

A best rank-(2,2,2) approximation always exists for order-3 tensors.

Simulations illustrate the theoretical criterion.

Abstract

It is well known that a best rank-$R$ approximation of order-3 tensors may not exist for $R\ge 2$. A best rank-$(R,R,R)$ approximation always exists, however, and is also a best rank-$R$ approximation when it has rank (at most) $R$. For $R=2$ and real order-3 tensors it is shown that a best rank-2 approximation is also a local minimum of the best rank-(2,2,2) approximation problem. This implies that if all rank-(2,2,2) minima have rank larger than 2, then a best rank-2 approximation does not exist. This provides an easy-to-check criterion for existence of a best rank-2 approximation. The result is illustrated by means of simulations.