329 RWTH Publication No: 47332        2011        IGPM329.pdf
TITLE An Introduction to Hierachical (H-) Rank and TT-Rank of Tensors with Examples
AUTHORS Lars Grasedyck, Wolfgang Hackbusch
ABSTRACT We review two similar concepts of hierarchical rank of tensors (which extend the matrix rank to higher order tensors): the TT-rank and the H-rank (hierarchical or H-Tucker rank). Based on this notion of rank, one can define a data-sparse representation of tensors involving O(dnk + dk3 ) data for order d tensors with mode sizes n and rank k. Simple examples underline the differences and similarities between the different formats and ranks. Finally, we derive rank bounds for tensors in one of the formats based on the ranks in the other format.
KEYWORDS hierarchical Tucker, tensor rank, tensor approximation, tensor train
DOI 10.2478/cmam-2011-0016
PUBLICATION Computational methods in applied mathematics
11 (2011) 3, p. 291-304