Learning from Frustration: Torsor CNNs on Graphs

Daiyuan Li, Shreya Arya, Robert Ghrist
Proceedings of the 4th (2025) and 3rd (2024) NeurIPS Workshops on Symmetry and Geometry in Neural Representations, PMLR 282:298-315, 2026.

Abstract

Most equivariant neural networks rely on a single global symmetry, limiting their use in domains where symmetries are instead local. We introduce Torsor CNNs, a framework for learning on graphs with local symmetries encoded as edge potentials—group-valued transformations between neighboring coordinate frames. We establish that this geometric construction is fundamentally equivalent to the classical group synchronization problem, yielding: (1) a Torsor Convolutional Layer that is provably equivariant to local changes in coordinate frames, and (2) the frustration loss—a standalone geometric regularizer that encourages locally equivariant representations when added to any NN’s training objective. The Torsor CNN framework unifies and generalizes several architectures—including classical CNNs and Gauge CNNs on manifolds—by operating on arbitrary graphs without requiring a global coordinate system or smooth manifold structure. We establish the mathematical foundations of this framework and demonstrate its applicability to multi-view 3D recognition, where relative camera poses naturally define the required edge potentials.

Cite this Paper

BibTeX
@InProceedings{pmlr-v282-li26a, title = {Learning from Frustration: Torsor CNNs on Graphs}, author = {Li, Daiyuan and Arya, Shreya and Ghrist, Robert}, booktitle = {Proceedings of the 4th (2025) and 3rd (2024) NeurIPS Workshops on Symmetry and Geometry in Neural Representations}, pages = {298--315}, year = {2026}, editor = {Acosta, Francisco and Azeglio, Simone and Tolooshams, Bahareh and van de Geijn, Chase and Shewmake, Christian and Sanborn, Sophia and Miolane, Nina}, volume = {282}, series = {Proceedings of Machine Learning Research}, month = {14 Dec 2024--07 Dec 2025}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v282/main/assets/li26a/li26a.pdf}, url = {https://proceedings.mlr.press/v282/li26a.html}, abstract = {Most equivariant neural networks rely on a single global symmetry, limiting their use in domains where symmetries are instead local. We introduce Torsor CNNs, a framework for learning on graphs with local symmetries encoded as edge potentials—group-valued transformations between neighboring coordinate frames. We establish that this geometric construction is fundamentally equivalent to the classical group synchronization problem, yielding: (1) a Torsor Convolutional Layer that is provably equivariant to local changes in coordinate frames, and (2) the frustration loss—a standalone geometric regularizer that encourages locally equivariant representations when added to any NN’s training objective. The Torsor CNN framework unifies and generalizes several architectures—including classical CNNs and Gauge CNNs on manifolds—by operating on arbitrary graphs without requiring a global coordinate system or smooth manifold structure. We establish the mathematical foundations of this framework and demonstrate its applicability to multi-view 3D recognition, where relative camera poses naturally define the required edge potentials.} }
Endnote
%0 Conference Paper %T Learning from Frustration: Torsor CNNs on Graphs %A Daiyuan Li %A Shreya Arya %A Robert Ghrist %B Proceedings of the 4th (2025) and 3rd (2024) NeurIPS Workshops on Symmetry and Geometry in Neural Representations %C Proceedings of Machine Learning Research %D 2026 %E Francisco Acosta %E Simone Azeglio %E Bahareh Tolooshams %E Chase van de Geijn %E Christian Shewmake %E Sophia Sanborn %E Nina Miolane %F pmlr-v282-li26a %I PMLR %P 298--315 %U https://proceedings.mlr.press/v282/li26a.html %V 282 %X Most equivariant neural networks rely on a single global symmetry, limiting their use in domains where symmetries are instead local. We introduce Torsor CNNs, a framework for learning on graphs with local symmetries encoded as edge potentials—group-valued transformations between neighboring coordinate frames. We establish that this geometric construction is fundamentally equivalent to the classical group synchronization problem, yielding: (1) a Torsor Convolutional Layer that is provably equivariant to local changes in coordinate frames, and (2) the frustration loss—a standalone geometric regularizer that encourages locally equivariant representations when added to any NN’s training objective. The Torsor CNN framework unifies and generalizes several architectures—including classical CNNs and Gauge CNNs on manifolds—by operating on arbitrary graphs without requiring a global coordinate system or smooth manifold structure. We establish the mathematical foundations of this framework and demonstrate its applicability to multi-view 3D recognition, where relative camera poses naturally define the required edge potentials.
APA
Li, D., Arya, S. & Ghrist, R.. (2026). Learning from Frustration: Torsor CNNs on Graphs. Proceedings of the 4th (2025) and 3rd (2024) NeurIPS Workshops on Symmetry and Geometry in Neural Representations, in Proceedings of Machine Learning Research 282:298-315 Available from https://proceedings.mlr.press/v282/li26a.html.

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