Computing the unique CANDECOMP/PARAFAC decomposition of unbalanced tensors by homotopy method

TL;DR

This paper establishes conditions for the unique CP decomposition of unbalanced tensors and introduces a homotopy-based algorithm to compute it, ensuring generic uniqueness under certain rank bounds.

Contribution

It derives a new upper bound for tensor rank guaranteeing generic uniqueness and develops a homotopy continuation algorithm for computing the CP decomposition.

Findings

Derived an upper bound for tensor rank ensuring uniqueness.

Developed a homotopy continuation algorithm for CP decomposition.

Proved the bound depends only on tensor dimensions.

Abstract

The Candecomp/Parafac (CP) decomposition of the tensor whose maximal dimension is greater than its rank is considered. We derive the upper bound of rank under which the generic uniqueness of CP decomposition is guaranteed. The bound only depends on the dimension of the tensor and the proof is constructive. Under these conditions, an algorithm applying homotopy continuation method is developed for computing the CP decomposition of tensors.