Kronecker product approximation with multiple factor matrices via the tensor product algorithm

摘要

Kronecker product (KP) approximation has recently been applied as a modeling and analysis tool on systems with hierarchical networked structure. In this paper, we propose a tensor product-based approach to the KP approximation problem with arbitrary number of factor matrices. The formulation involves a novel matrix-to-tensor transformation to convert the KP approximation problem to a best rank-(R1, ..., RN) tensor product approximation problem. In addition, we develop an algorithm based on higher-order orthogonal iteration to solve the tensor approximation problem. We prove that the proposed approach is equivalent to conventional singular value decomposition-based approach for two matrix factor case proposed by Van Loan. Hence, our work is a generalization of Van Loan's approach to more than two factor matrices. We demonstrate our approach by several experiments and case studies. The results indicate that the tensor product formulation is effective for KP approximation.