Containments of Tensor Network Varieties

TL;DR

This work generalizes the Hackbusch containment problem to tensor-network varieties $\mathrm{TNS}(\overline{\mathcal{T}})$ arising from full binary rooted trees, introducing the containment exponent $\mathcal{E}_{\mathcal{T},\mathcal{T}'}$ to quantify how scaling network parameters enables containment. It develops both inner (dimension-subspace nesting) and outer (contractions/DOAD sets) descriptions, and provides combinatorial (poset-based) and plane-tree bounds for these exponents, alongside a transitive framework based on comparing to the plane train-track tree $T\mathcal{T}_n$. The authors implement an algorithm to bound exponents for all trees with up to $n\le 8$ leaves and perform an exhaustive search over tree-label permutations, reporting results for small $n$ and making data available publicly; they also formulate an integer-programming approach to compute containment exponents for larger cases. Overall, the paper contributes a concrete, computable framework to compare tensor-network encodings, yields new theoretical bounds on containment, and opens questions about the tightness of these bounds and potential algorithmic enhancements. The results advance understanding of how tensor network representations relate across different tree-structured encodings and provide practical tools for quantifying these relationships.

Abstract

Building upon the work of Buczyńska 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.

Figures (6)

• Figure 1: Hierarchical tree $H\mathcal{T}_2$ of depth 2, with 4 leaves.
• Figure 2: Train track tree $T\mathcal{T}_4$ with 4 leaves.
• Figure 3: Network for matrices of rank $\le r$.
• Figure 4: Labelled binary tree $\mathcal{T}$.
• Figure 6: Two trees with 6 leaves.

Theorems & Definitions (34)

• Definition 2.1
• Definition 2.2
• Proposition 2.3
• Proposition 2.4
• Theorem 2.5: Hackbusch Conjecture, Theorem 4.3, BBM15
• Proposition 2.6: Proposition 4.6, BBM15
• Definition 2.7
• Proposition 2.8: Proposition 2.6, Lemma 2.8 BBM15
• Definition 2.9
• Lemma 2.10