# Learning from Frustration: Torsor CNNs on Graphs

arXiv: https://arxiv.org/abs/2510.23288 · HTML: https://mlsanity.com/paper/2510.23288/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 27 references that are in the map.
- First posted on arXiv 27 Oct 2025.

## Builds on (most cited first, 17 of 17)

- [FaceNet: A Unified Embedding for Face Recognition and Clustering](https://mlsanity.com/paper/1503.03832/) (2015)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [In Defense of the Triplet Loss for Person Re-Identification](https://mlsanity.com/paper/1703.07737/) (2017)
- [Multi-view Convolutional Neural Networks for 3D Shape Recognition](https://mlsanity.com/paper/1505.00880/) (2015)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [Harmonic Networks: Deep Translation and Rotation Equivariance](https://mlsanity.com/paper/1612.04642/) (2016)
- [Gauge Equivariant Convolutional Networks and the Icosahedral CNN](https://mlsanity.com/paper/1902.04615/) (2019)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [Neural Sheaf Diffusion: A Topological Perspective on Heterophily and Oversmoothing in GNNs](https://mlsanity.com/paper/2202.04579/) (2022)
- [Sheaf Neural Networks](https://mlsanity.com/paper/2012.06333/) (2020)
- [SE-Sync: A Certifiably Correct Algorithm for Synchronization over the Special Euclidean Group](https://mlsanity.com/paper/1612.07386/) (2016)
- [Equivariant Multi-View Networks](https://mlsanity.com/paper/1904.00993/) (2019)
- [N-body Networks: a Covariant Hierarchical Neural Network Architecture for Learning Atomic Potentials](https://mlsanity.com/paper/1803.01588/) (2018)
- [Homogeneous vector bundles and G-equivariant convolutional neural networks](https://mlsanity.com/paper/2105.05400/) (2021)
- [The Geometry of Synchronization Problems and Learning Group Actions](https://mlsanity.com/paper/1610.09051/) (2016)

## Built on by (most cited first, 2 of 2)

- [Consistent Geometric Deep Learning via Hilbert Bundles and Cellular Sheaves](https://mlsanity.com/paper/2605.06395/) (2026)
- [Equivariant Sheaf Neural Networks: Learning Geometric Transport on Graphs](https://mlsanity.com/paper/2608.28853/) (2026)

## Abstract (arXiv)

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.