A Geometric Insight into Equivariant Message Passing Neural Networks on Riemannian Manifolds

Ilyes Batatia
Proceedings of 2nd Annual Workshop on Topology, Algebra, and Geometry in Machine Learning (TAG-ML), PMLR 221:426-436, 2023.

Abstract

This work proposes a geometric insight into equivariant message passing on Riemannian manifolds. As previously proposed, numerical features on Riemannian manifolds are represented as coordinate-independent feature fields on the manifold. To any coordinate-independent feature field on a manifold comes attached an equivariant embedding of the principal bundle to the space of numerical features. We argue that the metric this embedding induces on the numerical feature space should optimally preserve the principal bundle’s original metric. This optimality criterion leads to the minimization of a twisted form of the Polyakov action with respect to the graph of this embedding, yielding an equivariant diffusion process on the associated vector bundle. We obtain a message passing scheme on the manifold by discretizing the diffusion equation flow for a fixed time step. We propose a higher- order equivariant diffusion process equivalent to diffusion on the cartesian product of the base manifold. The discretization of the higher-order diffusion process on a graph yields a new general class of equivariant GNN, generalizing the ACE and MACE formalism to data on Riemannian manifolds.

Cite this Paper

BibTeX
@InProceedings{pmlr-v221-batatia23a, title = {A Geometric Insight into Equivariant Message Passing Neural Networks on Riemannian Manifolds}, author = {Batatia, Ilyes}, booktitle = {Proceedings of 2nd Annual Workshop on Topology, Algebra, and Geometry in Machine Learning (TAG-ML)}, pages = {426--436}, year = {2023}, editor = {Doster, Timothy and Emerson, Tegan and Kvinge, Henry and Miolane, Nina and Papillon, Mathilde and Rieck, Bastian and Sanborn, Sophia}, volume = {221}, series = {Proceedings of Machine Learning Research}, month = {28 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v221/batatia23a/batatia23a.pdf}, url = {https://proceedings.mlr.press/v221/batatia23a.html}, abstract = {This work proposes a geometric insight into equivariant message passing on Riemannian manifolds. As previously proposed, numerical features on Riemannian manifolds are represented as coordinate-independent feature fields on the manifold. To any coordinate-independent feature field on a manifold comes attached an equivariant embedding of the principal bundle to the space of numerical features. We argue that the metric this embedding induces on the numerical feature space should optimally preserve the principal bundle’s original metric. This optimality criterion leads to the minimization of a twisted form of the Polyakov action with respect to the graph of this embedding, yielding an equivariant diffusion process on the associated vector bundle. We obtain a message passing scheme on the manifold by discretizing the diffusion equation flow for a fixed time step. We propose a higher- order equivariant diffusion process equivalent to diffusion on the cartesian product of the base manifold. The discretization of the higher-order diffusion process on a graph yields a new general class of equivariant GNN, generalizing the ACE and MACE formalism to data on Riemannian manifolds.} }
Endnote
%0 Conference Paper %T A Geometric Insight into Equivariant Message Passing Neural Networks on Riemannian Manifolds %A Ilyes Batatia %B Proceedings of 2nd Annual Workshop on Topology, Algebra, and Geometry in Machine Learning (TAG-ML) %C Proceedings of Machine Learning Research %D 2023 %E Timothy Doster %E Tegan Emerson %E Henry Kvinge %E Nina Miolane %E Mathilde Papillon %E Bastian Rieck %E Sophia Sanborn %F pmlr-v221-batatia23a %I PMLR %P 426--436 %U https://proceedings.mlr.press/v221/batatia23a.html %V 221 %X This work proposes a geometric insight into equivariant message passing on Riemannian manifolds. As previously proposed, numerical features on Riemannian manifolds are represented as coordinate-independent feature fields on the manifold. To any coordinate-independent feature field on a manifold comes attached an equivariant embedding of the principal bundle to the space of numerical features. We argue that the metric this embedding induces on the numerical feature space should optimally preserve the principal bundle’s original metric. This optimality criterion leads to the minimization of a twisted form of the Polyakov action with respect to the graph of this embedding, yielding an equivariant diffusion process on the associated vector bundle. We obtain a message passing scheme on the manifold by discretizing the diffusion equation flow for a fixed time step. We propose a higher- order equivariant diffusion process equivalent to diffusion on the cartesian product of the base manifold. The discretization of the higher-order diffusion process on a graph yields a new general class of equivariant GNN, generalizing the ACE and MACE formalism to data on Riemannian manifolds.
APA
Batatia, I.. (2023). A Geometric Insight into Equivariant Message Passing Neural Networks on Riemannian Manifolds. Proceedings of 2nd Annual Workshop on Topology, Algebra, and Geometry in Machine Learning (TAG-ML), in Proceedings of Machine Learning Research 221:426-436 Available from https://proceedings.mlr.press/v221/batatia23a.html.

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