Block Row Kronecker-Structured Linear Systems With a Low-Rank Tensor Solution - Hendrikx, Stijn; De Lathauwer, Lieven (Frontiers in Applied Mathematics and Statistics, 2022)
Several problems in compressed sensing and completion of higher-order tensors can be formulated as a structured linear system with a constrained tensor as the solution. In particular, we consider block row Kronecker-structured linear systems with a low-rank canonical polyadic decomposition, a low multilinear rank multilinear singular value decomposition or a low tensor train rank tensor train constrained solution. In this paper, we provide algorithms that serve as tools for finding such solutions for a large, higher-order data tensor, given Kronecker-structured linear combinations of its entries. Consistent with the literature on compressed sensing, the number of linear combinations of entries needed to find a constrained solution is far smaller than the corresponding total number of entries in the original tensor. We derive conditions under which a canonical polyadic decomposition, multilinear singular value decomposition or tensor train solution can be retrieved from this type of structured linear systems and also derive the corresponding generic conditions. Finally, we validate our algorithms by comparing them to related randomized tensor decomposition algorithms and by reconstructing a hyperspectral image from compressed measurements.