Ergodic and Subhomogeneous Dynamics in Hyperbolic Neural Networks

Nico Alvarado, Sebastian Burgos
Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, PMLR 300:2953-2961, 2026.

Abstract

We analyze the long term behavior of hyperbolic neural networks through subhomogeneous layer maps, focusing on stability, growth control, and robustness under stochastic perturbations. This work unifies the standard hyperbolic models via explicit isometries and M{ö}bius operations, allowing statements to be transported across representations without loss of geometric meaning. Within this model invariant view, we study iterated, noise perturbed transformations and develop an ergodic theoretic framework that characterizes their asymptotic behavior, including conditions that promote stability and convergence of averaged iterates. Beyond theory, these insights inform practical design choices for training procedures that remain well-behaved in the presence of noise and avoid unbounded parameter growth, thereby supporting more reliable use of hyperbolic representations in hierarchical and graph structured learning tasks.

Cite this Paper

BibTeX
@InProceedings{pmlr-v300-alvarado26a, title = { Ergodic and Subhomogeneous Dynamics in Hyperbolic Neural Networks }, author = {Alvarado, Nico and Burgos, Sebastian}, booktitle = {Proceedings of The 29th International Conference on Artificial Intelligence and Statistics}, pages = {2953--2961}, year = {2026}, editor = {Khan, Emtiyaz and Li, Yingzhen and Solin, Arno and Ramdas, Aaditya}, volume = {300}, series = {Proceedings of Machine Learning Research}, month = {02--05 May}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v300/main/assets/alvarado26a/alvarado26a.pdf}, url = {https://proceedings.mlr.press/v300/alvarado26a.html}, abstract = { We analyze the long term behavior of hyperbolic neural networks through subhomogeneous layer maps, focusing on stability, growth control, and robustness under stochastic perturbations. This work unifies the standard hyperbolic models via explicit isometries and M{ö}bius operations, allowing statements to be transported across representations without loss of geometric meaning. Within this model invariant view, we study iterated, noise perturbed transformations and develop an ergodic theoretic framework that characterizes their asymptotic behavior, including conditions that promote stability and convergence of averaged iterates. Beyond theory, these insights inform practical design choices for training procedures that remain well-behaved in the presence of noise and avoid unbounded parameter growth, thereby supporting more reliable use of hyperbolic representations in hierarchical and graph structured learning tasks. } }
Endnote
%0 Conference Paper %T Ergodic and Subhomogeneous Dynamics in Hyperbolic Neural Networks %A Nico Alvarado %A Sebastian Burgos %B Proceedings of The 29th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2026 %E Emtiyaz Khan %E Yingzhen Li %E Arno Solin %E Aaditya Ramdas %F pmlr-v300-alvarado26a %I PMLR %P 2953--2961 %U https://proceedings.mlr.press/v300/alvarado26a.html %V 300 %X We analyze the long term behavior of hyperbolic neural networks through subhomogeneous layer maps, focusing on stability, growth control, and robustness under stochastic perturbations. This work unifies the standard hyperbolic models via explicit isometries and M{ö}bius operations, allowing statements to be transported across representations without loss of geometric meaning. Within this model invariant view, we study iterated, noise perturbed transformations and develop an ergodic theoretic framework that characterizes their asymptotic behavior, including conditions that promote stability and convergence of averaged iterates. Beyond theory, these insights inform practical design choices for training procedures that remain well-behaved in the presence of noise and avoid unbounded parameter growth, thereby supporting more reliable use of hyperbolic representations in hierarchical and graph structured learning tasks.
APA
Alvarado, N. & Burgos, S.. (2026). Ergodic and Subhomogeneous Dynamics in Hyperbolic Neural Networks . Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 300:2953-2961 Available from https://proceedings.mlr.press/v300/alvarado26a.html.

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