This repository contains a Jupyter notebook that provides an interactive exploration of Equivariant Neural Networks on Homogeneous Spaces, with a specific focus on Spherical CNNs. The notebook complements Lecture 8 from the "Lie Groups with Applications" course through Quantum Formalism, demonstrating the deep connection between abstract Lie theory and modern geometric deep learning models.
- Homogeneous Spaces: Visualization of the sphere S² as the quotient space SO(3)/SO(2) through the orbit of points under group actions
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Feature Maps & Associated Vector Bundles: Interactive demonstration of how vector fields on S² can be understood as sections of the tangent bundle, in terms of an associated vector bundle:
$$TS^2 \cong SO(3) \times_{SO(2)} \mathbb{R}^2$$ . - Group Actions on Feature Fields: Implementation of the "push-pull" formula for SO(3) action on vector fields
- Spherical Harmonics: Exploration of spherical harmonics as an equivariant basis for functions on the sphere
- Harmonic Analysis & Spherical Harmonic Transform: Numerical representation of spherical functions through their harmonic coefficients
- Linear Equivariant Maps & Schur's Lemma: Demonstration of how representation theory constrains the structure of equivariant linear maps
- Convolution Equivalence: Verification of the fundamental theorem (Cohen et al., 2019) linking equivariant maps to spherical convolutions
- Practical Implementation with e3nn: Construction and testing of SO(3)-equivariant layers using the e3nn library
The notebook features several interactive visualizations:
- 3D rendering of spherical points and vector fields
- Group actions on geometric features
- Spherical harmonics as scalar fields on S²
- Equivariant vs. non-equivariant map outputs
This material builds on differential geometry, representation theory, and harmonic analysis:
- Lie Groups & Homogeneous Spaces: The sphere S² as SO(3)/SO(2)
- Induced Representations: How vector fields on S² relate to representations of SO(2)
- Spherical Harmonics: As irreducible representations of SO(3)
- Schur's Lemma: Application to equivariant maps between function spaces
- Convolution Theorem: The equivalence between equivariant maps and convolutions
These concepts form the backbone of equivariant deep learning models with applications in:
- Climate science and weather prediction
- 3D computer vision and point cloud processing
- Molecular modeling and drug discovery
- Astrophysics and cosmology (CMB analysis)
- Computer graphics and 3D shape analysis
- Python 3.8+
- NumPy, SciPy
- Matplotlib
- PyTorch
- e3nn (Equivariant Neural Network library)
This notebook can be run with standard Jupyter environments. Each code cell builds on previous sections, gradually constructing the full picture of equivariant neural networks on homogeneous spaces.
- Cohen, T. S., Geiger, M., & Weiler, M. (2019). A General Theory of Equivariant CNNs on Homogeneous Spaces. NeurIPS 32.
- Gerken, J.E., Aronsson, J., Carlsson, O. et al. Geometric deep learning and equivariant neural networks. Artif Intell Rev 56, 14605–14662 (2023).
Brian Hepler, PhD is a research mathematician exploring connections to data science, machine learning, and quantum computing. This material was developed as part of the "Lie Groups with Applications" course at Quantum Formalism Academy.
Copyright © 2025 Quantum Formalism Academy. All rights reserved. This repository is intended for educational purposes. Redistribution, modification, or commercial use of this material without prior written permission is prohibited.