日時： 2016年8月3日(水) 17:30–18:30
場所： 東京大学工学部 14号館 5階 534セミナー室
講演者：André Uschmajew (University of Bonn)
題目： Interconnections between higher-order singular values of real tensors
概要
The higher-order singular values for a tensor of order $d$ are defined
as the singular values of the $d$ different matricizations 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 qualititative 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).