Bilinear factorizations subject to monomial equality constraints via tensor decompositions

• Published in: Linear algebra and its applications Vol. 621; pp. 296 - 333
• Main Authors: Sørensen, Mikael, De Lathauwer, Lieven, Sidiropoulos, Nicholaos D.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 15.07.2021; American Elsevier Company, Inc
• Subjects: Algorithms; Block term decomposition; Canonical polyadic decomposition; Coupled decomposition; Decomposition; Eigenvalue decomposition; Eigenvalues; Equality; Factorization; Linear algebra; Machine learning; Mathematical analysis; Monomial; Signal processing; Tensor; Tensors; Uniqueness; Canonical polyadic decomposition; Tensor; Coupled decomposition; 15A23; Uniqueness; Monomial; Eigenvalue decomposition; 15A15; Block term decomposition
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2021.03.022

The Canonical Polyadic Decomposition (CPD), which decomposes a tensor into a sum of rank one terms, plays an important role in signal processing and machine learning. In this paper we extend the CPD framework to the more general case of bilinear factorizations subject to monomial equality constraints. This includes extensions of multilinear algebraic uniqueness conditions originally developed for the CPD. We obtain a deterministic uniqueness condition that admits a constructive interpretation. Computationally, we reduce the bilinear factorization problem into a CPD problem, which can be solved via a matrix EigenValue Decomposition (EVD). Under the given conditions, the discussed EVD-based algorithms are guaranteed to return the exact bilinear factorization. Finally, we make a connection between bilinear factorizations subject to monomial equality constraints and the coupled block term decomposition, which allows us to translate monomial structures into low-rank structures.