MatrixNet: Learning over symmetry groups using learned group representations

2025-01-16

AI Summary

Machine learning models often struggle to capture complex symmetries, limiting their generalization and sample efficiency in symmetry-sensitive tasks such as robotic manipulation and protein structure analysis. Method: We propose the first end-to-end differentiable framework for learning group matrix representations—without requiring predefined group representations—by directly mapping group elements to matrices that provably satisfy group relations. We introduce group relation-preserving regularization and word-length-aware training to ensure structural fidelity and enable reliable extrapolation to longer sequences. Contribution/Results: Our approach significantly outperforms standard equivariant baselines on finite group and Artin braid group prediction tasks, achieving higher accuracy, stronger zero-shot generalization to unseen group elements, and robust inference on sequences of previously unobserved lengths—demonstrating improved representational capacity for abstract algebraic structures in deep learning.

Technology Category

• Machine Learning: Representation Learning
• Computer Vision: Representation Learning for Vision
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Group theory has been used in machine learning to provide a theoretically grounded approach for incorporating known symmetry transformations in tasks from robotics to protein modeling. In these applications, equivariant neural networks use known symmetry groups with predefined representations to learn over geometric input data. We propose MatrixNet, a neural network architecture that learns matrix representations of group element inputs instead of using predefined representations. MatrixNet achieves higher sample efficiency and generalization over several standard baselines in prediction tasks over the several finite groups and the Artin braid group. We also show that MatrixNet respects group relations allowing generalization to group elements of greater word length than in the training set.

Problem

• Symmetry Learning
• Machine Learning Models
• Complex Symmetry Understanding

Innovation

• MatrixNet
• equivariant neural networks
• group symmetry learning