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

By Lars Grasedyck, Wolfgang Hackbusch. 2011

112 citations recorded by the source corpus.

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 + 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.

Subfields: Computational Mathematics, Computational Mechanics, Computer Networks and Communications

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Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “A discussion on the difference between the ranks for the HT and TT decompositions can be found in [105].”
DMRG+QTT approach to computation of the ground state for the molecular Schrödinger operator (2010, background): “For more detailed discussion on theoretical aspects of H-Tucker and TT formats, see [11, 7].”
LOCAL CONVERGENCE OF ALTERNATING SCHEMES FOR OPTIMIZATION OF CONVEX PROBLEMS IN THE TT FORMAT (2011, background): “[1, 2, 20, 24, 27]; while the former approach seems to be superior on some problems in the sense that it sometimes provides lower ranks [15], the latter has advantage of being conceptually quite simple, thus often allowing for a simpler analysis from the theoretical point of view, and enabling, from”
The geometry of algorithms using hierarchical tensors (2013, background): “Much of this and related concepts were already introduced in [11,12,52] but we establish in addition some new relations on the parametrization of this set.”
Tensor product methods and entanglement optimization for ab initio quantum chemistry (2014, unknown): “α2=1 A2(m1, α2,m{1,2}) |ξ{1} m1 〉 ⊗ |φ{2} α2 〉 (77)”
On Local Convergence of Alternating Schemes for Optimization of Convex Problems in the Tensor Train Format (2013, background): “[1, 2, 22, 26, 29]; while the former approach seems to be superior on some problems in the sense that it sometimes provides lower ranks [16], the latter has advantage of being conceptually quite simple, thus often allowing for a simpler analysis from the theoretical point of view, and enabling, from”
Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 1 Low-Rank Tensor Decompositions (2016, unknown): “, no edges connecting a node with itself; also, the splitting of the modes of the original data tensor by binary tree edges is performed through a suitable matricization [107, 108].”
Model Compression and Hardware Acceleration for Neural Networks: A Comprehensive Survey (2020, unknown): “[132] L.”