Linear Systems with a Canonical Polyadic Decomposition Constrained Solution: Algorithms and Applications

© 2018 John Wiley & Sons, Ltd. Real-life data often exhibit some structure and/or sparsity, allowing one to use parsimonious models for compact representation and approximation. When considering matrix and tensor data, low-rank models such as the (multilinear) singular value decomposition, canonical polyadic decomposition (CPD), tensor train, and hierarchical Tucker model are very common. The solution of (large-scale) linear systems is often structured in a similar way, allowing one to use compact matrix and tensor models as well. In this paper, we focus on linear systems with a CPD-constrained solution (LS-CPD). Our main contribution is the development of optimization-based and algebraic methods to solve LS-CPDs. Furthermore, we propose a condition that guarantees generic uniqueness of the obtained solution. We also show that LS-CPDs provide a broad framework for the analysis of multilinear systems of equations. The latter are a higher-order generalization of linear systems, similar to tensor decompositions being a generalization of matrix decompositions. The wide applicability of LS-CPDs in domains such as classification, multilinear algebra, and signal processing is illustrated.status: publishe