AUTHOR=Hendrikx Stijn , De Lathauwer Lieven TITLE=Block Row Kronecker-Structured Linear Systems With a Low-Rank Tensor Solution JOURNAL=Frontiers in Applied Mathematics and Statistics VOLUME=8 YEAR=2022 URL=https://www.frontiersin.org/journals/applied-mathematics-and-statistics/articles/10.3389/fams.2022.832883 DOI=10.3389/fams.2022.832883 ISSN=2297-4687 ABSTRACT=

Several problems in compressed sensing and randomized tensor decomposition 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 multilinear rank multilinear singular value decomposition, a low-rank canonical polyadic 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 multilinear singular value decomposition, canonical polyadic 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.