Generic uniqueness conditions for the canonical polyadic decomposition and INDSCAL

TL;DR

This paper establishes new generic conditions ensuring the uniqueness of the canonical polyadic decomposition (CPD) of third-order tensors and extends these results to the INDSCAL model with symmetric slices, using algebraic geometry.

Contribution

It provides improved and relaxed generic uniqueness conditions for CPD and INDSCAL, enhancing theoretical understanding in tensor decomposition.

Findings

New conditions guarantee CPD uniqueness for generic tensors.

Relaxed bounds for INDSCAL uniqueness with symmetric slices.

Algebraic geometry techniques underpin the proofs.

Abstract

We find conditions that guarantee that a decomposition of a generic third-order tensor in a minimal number of rank-$1$ tensors (canonical polyadic decomposition (CPD)) is unique up to permutation of rank-$1$ tensors. Then we consider the case when the tensor and all its rank-$1$ terms have symmetric frontal slices (INDSCAL). Our results complement the existing bounds for generic uniqueness of the CPD and relax the existing bounds for INDSCAL. The derivation makes use of algebraic geometry. We stress the power of the underlying concepts for proving generic properties in mathematical engineering.