Automatic Discovery of One Parameter Subgroups of $SO(n)$

2025-09-26

AI Summary

Existing methods lack automated discovery of one-parameter subgroups of SO(n). Method: This paper proposes an interpretable learning framework grounded in Lie algebra theory and the Jordan canonical form of skew-symmetric matrices: it first establishes a canonical representation of group orbits, then constructs families of invariant functions to enable parametric modeling of continuous symmetries. Contribution/Results: To our knowledge, this is the first approach to end-to-end, data-driven identification of SO(n) one-parameter subgroups and their symmetry-aware functional representations. Evaluated on diverse tasks—including double-pendulum dynamics modeling, moment-of-inertia prediction, top-quark tagging, and invariant polynomial regression—the model successfully recovers physically or geometrically meaningful subgroup structures. It achieves high generalization performance while delivering human-interpretable symmetry characterizations, bridging the gap between learned representations and domain-specific invariances.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Learning to Search
• Intelligent Robots: Learning & Optimization for ROB

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

We introduce a novel framework for the automatic discovery of one-parameter subgroups ($H_γ$) of $SO(3)$ and, more generally, $SO(n)$. One-parameter subgroups of $SO(n)$ are crucial in a wide range of applications, including robotics, quantum mechanics, and molecular structure analysis. Our method utilizes the standard Jordan form of skew-symmetric matrices, which define the Lie algebra of $SO(n)$, to establish a canonical form for orbits under the action of $H_γ$. This canonical form is then employed to derive a standardized representation for $H_γ$-invariant functions. By learning the appropriate parameters, the framework uncovers the underlying one-parameter subgroup $H_γ$. The effectiveness of the proposed approach is demonstrated through tasks such as double pendulum modeling, moment of inertia prediction, top quark tagging and invariant polynomial regression, where it successfully recovers meaningful subgroup structure and produces interpretable, symmetry-aware representations.

Problem

• Automatically discovers one-parameter subgroups of SO(n)
• Establishes canonical forms for orbits and invariant functions
• Learns subgroup parameters for symmetry-aware representations

Innovation

• Using Jordan form for canonical orbits
• Standardizing representation of invariant functions
• Learning parameters to discover subgroups