On the best low multilinear rank approximation of higher-order tensors

This paper deals with the best low multilinear rank approximation of higher-order tensors. Given a tensor, we are looking for another tensor, as close as possible to the given one and with bounded multilinear rank. Higher-order tensors are used in higher-order statistics, signal processing, telecommunications and many other fields. In particular, the best low multilinear rank approximation is used as a tool for dimensionality reduction and signal subspace estimation. Higher-order generalizations of the singular value decomposition exist but lead to suboptimal solutions of the problem. The higher-order orthogonal iteration is an iterative algorithm for further refinement. It has linear conver gence speed. We aim for conceptually faster algorithms. However, there are infinitely many equivalent solutions whereas standard optimization algorithms have nice convergence properties if the solutions are isolated. The present invariance can be removed by working on quotient matrix manifolds. We discuss three algorithms, based on Newton's method, on the trust-region scheme and on conjugate gradients. We also comment on the local minima of the problem.