On the Uniqueness of the Canonical Polyadic Decomposition of third-order tensors --- Part I: Basic Results and Uniqueness of One Factor Matrix

TL;DR

This paper reviews and extends conditions for the uniqueness of the Canonical Polyadic Decomposition of third-order tensors, focusing on the case where one factor matrix's uniqueness is guaranteed by new, relaxed criteria involving compound matrices.

Contribution

It introduces new, relaxed conditions involving Khatri-Rao products of compound matrices that ensure the uniqueness of one factor matrix in CPD, and provides a shorter proof for cases with full column rank.

Findings

New conditions guarantee uniqueness of one factor matrix.

Shorter proof for CPD uniqueness with full column rank factor matrix.

Development of foundational material on compound matrices for future work.

Abstract

Canonical Polyadic Decomposition (CPD) of a higher-order tensor is decomposition in a minimal number of rank-1 tensors. We give an overview of existing results concerning uniqueness. We present new, relaxed, conditions that guarantee uniqueness of one factor matrix. These conditions involve Khatri-Rao products of compound matrices. We make links with existing results involving ranks and k-ranks of factor matrices. We give a shorter proof, based on properties of second compound matrices, of existing results concerning overall CPD uniqueness in the case where one factor matrix has full column rank. We develop basic material involving $m$-th compound matrices that will be instrumental in Part II for establishing overall CPD uniqueness in cases where none of the factor matrices has full column rank.