Singular value decompositions and low rank approximations of multi-linear functionals

The singular value decomposition is among the most important algebraic tools for solving many approximation problems in model reduction, data compression, system identification and signal processing. Nevertheless, there is no straightforward generalization of the algebraic concept of singular values and singular value decompositions to multi-linear functions. Motivated by the problem of finding lower rank approximations of tensors, this paper introduces a notion of singular values for arbitrary multi-linear mappings. An upperbound is derived on the error between a tensor and its optimal lower rank approximation and a conceptual algorithm is proposed to compute singular value decompositions of tensors.