Machine learning and invariant theory

Abstract

Inspired by constraints from physical law, equivariant machine learning restricts the learning to a hypothesis class where all the functions are equivariant with respect to some group action. Irreducible representations or invariant theory are typically used to parameterize the space of such functions. In this article, we introduce the topic and explain a couple of methods to explicitly parameterize equivariant functions that are being used in machine learning applications. In particular, we explicate a general procedure, attributed to Malgrange, to express all polynomial maps between linear spaces that are equivariant under the action of a group $G$, given a characterization of the invariant polynomials on a bigger space. The method also parametrizes smooth equivariant maps in the case that $G$ is a compact Lie group.

References in corpus (28)

• Neural Message Passing for Quantum Chemistry
• Spectral Networks and Locally Connected Networks on Graphs
• E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials
• Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds
• SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks
• Gauge Equivariant Convolutional Networks and the Icosahedral CNN
• GemNet: Universal Directional Graph Neural Networks for Molecules
• e3nn: Euclidean Neural Networks
• Equivariance Through Parameter-Sharing
• EquiBind: Geometric Deep Learning for Drug Binding Structure Prediction
• A numerical algorithm for the explicit calculation of SU(N) and SL(N,C) Clebsch-Gordan coefficients
• Incorporating Symmetry into Deep Dynamics Models for Improved Generalization
• Lorentz Group Equivariant Neural Network for Particle Physics
• Relevance of Rotationally Equivariant Convolutions for Predicting Molecular Properties
• Scalars are universal: Equivariant machine learning, structured like classical physics
• Sign and Basis Invariant Networks for Spectral Graph Representation Learning
• Learning with invariances in random features and kernel models
• On the Universality of Rotation Equivariant Point Cloud Networks
• Dimensionless machine learning: Imposing exact units equivariance
• Approximately Equivariant Networks for Imperfectly Symmetric Dynamics
• Towards fully covariant machine learning
• Equivariance versus Augmentation for Spherical Images
• Data Augmentation vs. Equivariant Networks: A Theory of Generalization on Dynamics Forecasting
• Low Dimensional Invariant Embeddings for Universal Geometric Learning
• Group-invariant max filtering
• A new algorithm for computing branching rules and Clebsch-Gordan coefficients of unitary representations of compact groups
• Invariant polynomials and machine learning
• A simple equivariant machine learning method for dynamics based on scalars