The Hackbusch conjecture on tensor formats - part two

TL;DR

This paper proves a generalized version of Hackbusch's conjecture, providing an algorithm to compute flattening ranks in tensor network models and showing exponential growth of tensor rank with the number of leaves.

Contribution

It extends Hackbusch's conjecture to more general tensor network models and introduces an algorithm for flattening rank computation in these models.

Findings

Algorithm for flattening rank computation in TNS models

Exponential growth of tensor rank with number of leaves

Generalization of Hackbusch's conjecture to broader tensor formats

Abstract

We prove a conjecture of W.~Hackbusch in a bigger generality than in our previous article. Here we consider Tensor Train (TT) model with an arbitrary number of leaves and a corresponding "almost binary tree" for Hierarchical Tucker (HT) model, i.e. the deepest tree with the same number of leaves. Our main result is an algorithm that computes the flattening rank of a generic tensor in a Tensor Network State (TNS) model on a given tree with respect to any flattening coming from combinatorics of the space. The methods also imply that the tensor rank (which is also called CP-rank) of most tensors in a TNS model grows exponentially with the growth of the number of leaves for any shape of the tree.