The Hackbusch conjecture on tensor formats - part two

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.