Multi-set low-rank factorizations with shared and unshared components

Low-rank matrix/tensor factorizations play a significant role in science and engineering. An important example is the canonical polyadic decomposition (CPD). There is also a growing interest in multi-set extensions of low-rank matrix/tensor factorizations in which the associated factor matrices are partially shared. In this paper we propose a more unified framework for multi-set matrix/tensor factorizations. In particular, we propose a multi-set extension of bilinear factorizations subject to monomial equality constraints to the case of shared and unshared factors. The presented framework encompasses (generalized) canonical correlation analysis (CCA) and (coupled) CPD models as special cases. CPD, CCA and hybrid models between them feature interesting uniqueness properties. We derive uniqueness conditions for CCA and multi-set low-rank factorization with partially shared entities. Computationally, we reduce multi-set low-rank factorizations with shared and unshared components into a special CPD problem, which can be solved via a matrix eigenvalue decomposition. Finally, numerical experiments demonstrate the importance of taking the coupling between multi-set low-rank factorizations into account in the actual computation.