Hackbusch Conjecture on tensor formats

TL;DR

This paper proves Hackbusch's conjecture comparing the complexities of tensor network state encodings between hierarchical and tensor train formats, using binary trees to represent tensor structures.

Contribution

It establishes a formal proof relating the complexity differences between hierarchical and tensor train tensor formats for specific binary tree structures.

Findings

Proves Hackbusch's conjecture on tensor format complexities.

Shows the relationship between hierarchical and tensor train formats.

Provides insights into tensor network state representations.

Abstract

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.