The paper applies parameterised graph theory to tensor networks, showing that cutwidth and tree‑cutwidth bound the bond‑dimension overhead needed to represent a tensor‑network state as a matrix product state or tree tensor network. It derives graph‑dependent upper bounds on the sample and computational complexity of tensor‑network tomography, introducing a new graph parameter called learning complexity. Finally, it extends the framework to an agnostic learner that approximates any state with a tensor‑network state of given bond dimension, providing explicit graph‑dependent complexity bounds.
By Matthias C. Caro, Natalie McHugh, Sergii Strelchuk
The paper presents a unified theoretical framework for analyzing and constructing deep neural networks by explicitly modeling tensor operations, which are often abstracted in existing theories. This framework allows researchers to study how architectural complexity has evolved over the past 40 years and to automatically generate new architectures using novel tensor operations. The authors also release a dataset of over 3,000 higher‑complexity architectures that have not yet been explored, providing a resource for future research.
By Nicholas J. Cooper, Fran\c{c}ois G. Meyer, Michael L. Roberts, Carlos Zapata-Carratal\'a, Lijun Chen, Danna Gurari
We investigate message-passing graph neural networks with random node features. Random node features are known to enhance the expressiveness of graph neural networks (GNNs) both theoretically and empirically.
arXiv:2608.29279v1 Announce Type: new
Abstract: Deep learning is often criticized for its theoretical research lagging behind practice. To make deep learning easier to understand, the entropy space t...
By Li Li, Tong Zhang, Wentao Yu, Zuobin Wang
arXiv:2608. 01357v1 Announce Type: new Abstract: Traditional approximation theory measures convergence rates in terms of the number of parameters or degrees of freedom.
By Tong Mao, Jinchao Xu
The paper presents a formal analysis of the quotient geometry of tree tensor networks (TTNs) and introduces efficient first- and second-order optimization algorithms that leverage this geometry. It also develops a backpropagation method for training TTNs in a kernel learning context. Numerical experiments on a digit classification task demonstrate a tradeoff between two horizontal distributions: one provides clearer geometric insights, while the other yields more efficient algorithms.
By Marius Willner, Marco Trenti, Dirk Lebiedz
arXiv:2607. 26699v1 Announce Type: new Abstract: We investigate message-passing graph neural networks with random node features.
By Lukas Gonon, Thilo Meyer-Brandis, Niklas Weber
arXiv:2609.23774v1 Announce Type: new
Abstract: Probabilistic inference is generally only tractable in low-treewidth graphical models, limiting its effective applicability in high-treewidth settings....
By Sagad Hamid, Tanya Braun
The paper investigates how continuous random node features affect the expressivity of message‑passing neural networks (MPNNs). It introduces a population‑level framework for size‑uniform approximation, showing that continuous random features can be quantized to finitely many colors with negligible error, allowing transfer of finite‑color distinguishability limits to size‑uniform approximation. The authors prove that such MPNNs are not size‑uniformly universal, analyze expressivity gains from colors and independent trials, establish incomparability with higher‑order Weisfeiler‑Leman methods, and provide finite‑color MPNN constructions for tree subgraph isomorphism and a local minimum spanning tree task.
By Robert R. Nerem, Mitchell Black, Antonis Vasileiou, Christopher Morris, Yusu Wang
arXiv:2605.16610v2 Announce Type: replace
Abstract: High-dimensional data arise naturally in many areas of science and engineering, including machine learning, signal processing, computational physic...
By Beheshteh T. Rakhshan, Guillaume Rabusseau
arXiv:2510. 10101v4 Announce Type: replace Abstract: Understanding the interplay between generalization, expressivity, and the geometry of the input space is a central challenge in graph learning.
By Martin Carrasco, Caio F. Deberaldini Netto, Vahan A. Martirosyan, Ehimare Okoyomon, Caterina Graziani
arXiv:2608. 17135v1 Announce Type: cross Abstract: Tensor networks are powerful formats for compressing large-scale data.
By Xiao Wang, Tomohiro Hashizume, Pia Siegl, Dieter Jaksch