Diagonalization and low-rank appromixation of tensors: a singular value decomposition approach

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 multilinear functions. Motivated by the problem of finding lower rank approximations of tensors, this paper discusses different options for rank decompositions of tensors. Using modal rank decompositions some approximation properties are derived, both for arbitrary and diagonalizable tensors