Containments of Tensor Network Varieties

TL;DR

This paper investigates the containment relationships between tensor network varieties, introducing the containment exponent as a measure and providing algorithms and experiments to understand these relationships.

Contribution

It proposes a general framework for tensor network containments, introduces the containment exponent, and develops algorithms with experimental validation.

Findings

Defined the containment exponent for tensor network varieties.

Developed an algorithm to bound containment exponents.

Conducted an exhaustive search for trees with up to 8 leaves.

Abstract

Building upon the work of Buczy\'nska et al., we study here tensor formats and their corresponding encoding of tensors via two-fold tensor products determined by the combinatorics of a binary tree. The set of all tensors representable by a given network forms the corresponding tensor network variety. A very basic question asks whether every tensor representable by one network is representable by another network, namely, when one tensor network variety is contained in another. Specific instances of this question became known as the Hackbusch Conjecture. Here, we propose a general framework for this question and take first steps, theoretical as well as experimental, towards a better understanding. In particular, given any two binary trees on $n$ leaves, we define (and prove existence of) a new measure, the containment exponent, which gauges how much one has to boost the parameters of one network for the containment to hold. We present an algorithm for bounding these containment exponents of tensor network varieties and report on an exhaustive search among trees on up to $n=8$ leaves.