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Learning from Frustration: Torsor CNNs on Graphs

Daiyuan Li, Shreya Arya, Robert Ghrist

TL;DR

This paper addresses learning on graphs with local coordinate systems by introducing Torsor CNNs, which encode local frame changes via edge potentials $\psi_{uv}$ and yield gauge-equivariant layers. It proves an equivalence between local symmetry learning and the classical group synchronization problem, and introduces two practical tools: Torsor Convolutional Layers and the frustration loss, a geometric regularizer that promotes locally consistent representations. The framework generalizes existing architectures (CNNs, G-CNNs, Gauge CNNs) to arbitrary graphs without requiring global coordinates, and is demonstrated in multi-view 3D recognition where relative camera poses define the edge potentials. By unifying discrete geometry and deep learning, the method offers principled handling of local symmetries with potential improvements in data efficiency and generalization, especially in domains with heterogeneous local frames. The practical validation on a multi-view dataset showcases the approach's applicability and potential to enhance performance with geometric regularization or fully gauge-equivariant layers.

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.

Learning from Frustration: Torsor CNNs on Graphs

TL;DR

This paper addresses learning on graphs with local coordinate systems by introducing Torsor CNNs, which encode local frame changes via edge potentials and yield gauge-equivariant layers. It proves an equivalence between local symmetry learning and the classical group synchronization problem, and introduces two practical tools: Torsor Convolutional Layers and the frustration loss, a geometric regularizer that promotes locally consistent representations. The framework generalizes existing architectures (CNNs, G-CNNs, Gauge CNNs) to arbitrary graphs without requiring global coordinates, and is demonstrated in multi-view 3D recognition where relative camera poses define the edge potentials. By unifying discrete geometry and deep learning, the method offers principled handling of local symmetries with potential improvements in data efficiency and generalization, especially in domains with heterogeneous local frames. The practical validation on a multi-view dataset showcases the approach's applicability and potential to enhance performance with geometric regularization or fully gauge-equivariant layers.

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.
Paper Structure (28 sections, 5 theorems, 25 equations, 1 figure)

This paper contains 28 sections, 5 theorems, 25 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our Approach.
  3. Connection to Synchronization.
  4. Generality.
  5. Applications.
  6. Contributions:
  7. Related Works: From Global to Local Equivariance
  8. G-CNNs on Homogeneous Spaces
  9. Gauge Equivariant CNNs on Manifolds
  10. Mathematical Background
  11. The Synchronization Problem
  12. Network Sheaves and Global Sections
  13. Network Torsors from Edge Potentials
  14. Gauge and Gauge Transformations
  15. Torsor CNNs on Graphs
  16. ...and 13 more sections

Key Result

Proposition 4.3

Given an edge potential $\psi$, there is a canonical bijection between feature fields $\Gamma(X, \mathcal{E})$ and functions $f:V \to F$ satisfying the feature synchronization condition: The bijection maps a section $\sigma$ to its representation $f$ in the identity gauge via $\sigma_v = [1_G, f_v]$, where $1_G \in G$ is the identity element. Moreover this correspondence is gauge-invariant: under

Figures (1)

  • Figure 1: (Left) Edge $e = (u,v)$ with edge potential $\psi_e$ (or sometimes written as $\psi_{uv}$) mapping between local frames. (Right) Associated vector sheaf construction where each torsor stalk (dotted line) carries feature vector spaces at different reference points. Global section (blue) corresponds to synchronized features satisfying $f_u = \rho(\psi_{uv})f_v$ across the edge.

Theorems & Definitions (27)

  • Definition 3.1
  • Example 3.2: Planar Rotation Synchronization
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • Definition 3.6
  • Definition 3.7
  • Definition 3.8
  • Definition 3.10
  • Definition 4.1
  • ...and 17 more