Publikation: The Hackbusch conjecture on tensor formats
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Zusammenfassung
We prove a conjecture of W. Hackbusch about tensor network states related to a perfect binary tree and train track tree. Tensor network states are used to present seemingly complicated tensors in a relatively simple and efficient manner. Each such presentation is described by a binary tree and a collection of vector spaces, one for each vertex of the tree. A problem suggested by Wolfgang Hackbusch and Joseph Landsberg is to compare the complexities of encodings, if one presents the same tensor with respect to two different trees. We answer this question when the two trees are extremal cases: the most "spread" tree (perfect binary tree), and the "deepest" binary tree (train track tree). The corresponding tensor formats are called hierarchical formats (HF) and tensor train (TT) formats, respectively.
Zusammenfassung in einer weiteren Sprache
On démontre une conjecture de W. Hackbusch concernant des réseaux de tenseurs associés à certains arbres. Les réseaux de tenseurs sont utilisés pour présenter des tenseurs apparemment compliqués d'une manière relativement simple et efficace. Chaque présentation d'un tenseur donné est décrite par un arbre binaire et une collection d'espaces vectoriels (un pour chaque sommet de l'arbre). Un problème posé par Wolfgang Hackbusch et Joseph Landsberg est de comparer les complexités des codages lorsque l'on présente le même tenseur via deux arbres différents. On répond à cette question lorsque les deux arbres sont des cas extrêmes : l'arbre binaire parfait d'une part et l'arbre binaire le plus « profond » d'autre part. Les formats des tenseurs correspondants sont appelés formats hiérarchiques (HF) et formats train de tenseurs (TT), respectivement.
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BUCZYŃSKA, Weronika, Jarosław BUCZYŃSKI, Mateusz MICHALEK, 2015. The Hackbusch conjecture on tensor formats. In: Journal de Mathématiques Pures et Appliquées. Elsevier. 2015, 104(4), pp. 749-761. ISSN 0021-7824. eISSN 1776-3371. Available under: doi: 10.1016/j.matpur.2015.05.002BibTex
@article{Buczynska2015Hackb-52351,
year={2015},
doi={10.1016/j.matpur.2015.05.002},
title={The Hackbusch conjecture on tensor formats},
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journal={Journal de Mathématiques Pures et Appliquées},
pages={749--761},
author={Buczyńska, Weronika and Buczyński, Jarosław and Michalek, Mateusz}
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