New Uniqueness Conditions for the Canonical Polyadic Decomposition of Third-Order Tensors

Abstract

The uniqueness properties of the canonical polyadic decomposition (CPD) of higher-order tensors make it an attractive tool for signal separation. However, CPD uniqueness is not yet fully understood. In this paper, we first present a new uniqueness condition for a polyadic decomposition (PD) where one of the factor matrices is assumed to be known. We also show that this result can be used to obtain a new overall uniqueness condition for the CPD. In signal processing the CPD factor matrices are often constrained. Building on the preceding results, we provide a new uniqueness condition for a CPD with a columnwise orthonormal factor matrix, representing uncorrelated signals. We also obtain a new uniqueness condition for a CPD with a partial Hermitian symmetry, useful for tensors in which covariance matrices are stacked, which are common in statistical signal processing. We explain that such constraints can lead to more relaxed uniqueness conditions. Finally, we provide an inexpensive algorithm for computing a PD with a known factor matrix that is also useful for the computation of the full CPD.