Singular Value Decompositions and Low Rank Approximations of Tensors

The singular value decomposition is among the most important tools in numerical analysis for solving a wide scope of approximation problems in signal processing, model reduction, system identification and data compression. Nevertheless, there is no straightforward generalization of the algebraic concepts underlying the classical singular values and singular value decompositions to multilinear functions. Motivated by the problem of lower rank approximations of tensors, this paper develops a notion of singular values for arbitrary multilinear mappings. We provide bounds on the error between a tensor and its optimal lower rank approximation. Conceptual algorithms are proposed to compute singular value decompositions of tensors.