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Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

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Volume 11, Issue 3 (Jan 2011)


An Introduction to Hierarchical (H-) Rank and TT-Rank of Tensors with Examples

Lars Grasedyck
  • Institut für Geometrie und Praktische Mathematik, RWTH Aachen, Templergraben 55, 52056 Aachen, Germany.
  • Email:
Wolfgang Hackbusch
  • Max Planck Institute for Mathematics in the Sciences, Inselstraße 22–26, 04103 Leipzig, Germany.
  • Email:


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 + dk^3) 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

About the article

Received: 2011-05-10

Revised: 2011-07-16

Accepted: 2011-08-20

Published in Print:

Citation Information: Computational Methods in Applied Mathematics Comput. Methods Appl. Math., ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.2478/cmam-2011-0016. Export Citation

© Institute of Mathematics, NAS of Belarus. This article is distributed under the terms of the Creative Commons Attribution Non-Commercial License, which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY-NC-ND 4.0)

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