André Uschmajew: Interconnections between higher-order singular values of real tensors

Abstract:

The higher-order singular values for a tensor of order d are defined as the singular values of the d different matricizations (flattenings) associated with the multilinear rank. They are important measures for low-rank approximability of the tensor in the Tucker format. It is therefore an interesting and important task to study the singular value vectors for different matricizations of a tensor simultaneously. The interconnections between the different modes are difficult to characterize quantitatively, but some qualitative questions can be addressed. One can for instance show that not all configurations of singular values are feasible in the sense that they can be realized by some tensor. One of our main results is that for generic tensors of order at least three and identical mode sizes, any small perturbation of the higher-order singular values remain feasible (up to a trivial norm constraint). This is in contrast to matrices, where the positive singular values of a matrix and its transpose are always the same. In order to construct a tensor with prescribed singular values, or even with prescribed Gramians, for different matricizations, a Newton-type method can be used as will be demonstrated by some numerical results.

This is joint work with Wolfgang Hackbusch (Leipzig) and Daniel Kressner (Lausanne).