Abstract of Ivan Oseledets

Tensor train and QTT decompositions for high-dimensional tensors
(joint with E.E. Tyrtyshnikov and B.N. Khoromskij) In this talk basic idea of tensor-train decomposition, which can be considered as natural extention of singular value decomposition to high dimensions. It does not suffer from the curse of dimensionality, and can be computed with the reliability and SVD. Basic subroutines are simple to implement and are available online. QTT decomposition opens a new application area for tensor decompositions --- approximation of tensors of "physically small" dimension. It includes compact representation of functions on sufficiently fine tensor grids with 2^D points in each direction, leading to d log n complexity. When the tensor is in structured format, it is interesting to perform some operations with it. Some operations are very intuitive in the tensor-train format, however some are not. An important operation is finding maximal and minimal elements. An algorithm usin maximal-volume submatrices will be presented for finding maximal in modulus element in the TT format.


Lars Grasedyck (MPI Leipzig, Germany)
Wolfgang Hackbusch (MPI Leipzig, Germany)
Boris Khoromskij (MPI Leipzig, Germany)