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Automatic Discovery of One-Parameter Subgroups of Lie Groups: Compact and Non-Compact Cases of $\mathbf{SO(n)}$ and $\mathbf{SL(n)}$

Pavan Karjol, Vivek V Kashyap, Rohan Kashyap, Prathosh A P

TL;DR

The paper tackles the problem of learning unknown rotational symmetries by automatically discovering one-parameter subgroups of $SO(n)$ and constructing symmetry-aware representations. It introduces $H_\gamma^{\text{inv}}$-Net, which jointly learns the subgroup parameters (orientation $A$ and rates $\{\lambda_i\}$) and an invariant function via a canonical representation invRep, unifying symmetry discovery with learning. Theoretical results provide canonical forms for $H_\gamma$-invariant functions in $SO(3)$ and generalize to $SO(n)$, with invRep establishing a bijection with orbits, enabling a simple, end-to-end approach. Empirical results on synthetic polynomials and real-world tasks (including quantum systems, double pendulum, inertia prediction, and Lorentz-related top quark tagging) show that the method accurately recovers the underlying rotation rates (near zero cosine distance to ground truth) and outperforms existing symmetry-discovery baselines. The work demonstrates broad applicability and interpretability of symmetry-aware models, and points to extensions to Lorentz groups and higher-order tensors as future directions.

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.

Automatic Discovery of One-Parameter Subgroups of Lie Groups: Compact and Non-Compact Cases of $\mathbf{SO(n)}$ and $\mathbf{SL(n)}$

TL;DR

The paper tackles the problem of learning unknown rotational symmetries by automatically discovering one-parameter subgroups of and constructing symmetry-aware representations. It introduces -Net, which jointly learns the subgroup parameters (orientation and rates ) and an invariant function via a canonical representation invRep, unifying symmetry discovery with learning. Theoretical results provide canonical forms for -invariant functions in and generalize to , with invRep establishing a bijection with orbits, enabling a simple, end-to-end approach. Empirical results on synthetic polynomials and real-world tasks (including quantum systems, double pendulum, inertia prediction, and Lorentz-related top quark tagging) show that the method accurately recovers the underlying rotation rates (near zero cosine distance to ground truth) and outperforms existing symmetry-discovery baselines. The work demonstrates broad applicability and interpretability of symmetry-aware models, and points to extensions to Lorentz groups and higher-order tensors as future directions.

Abstract

We introduce a novel framework for the automatic discovery of one-parameter subgroups () of and, more generally, . One-parameter subgroups of 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 , to establish a canonical form for orbits under the action of . This canonical form is then employed to derive a standardized representation for -invariant functions. By learning the appropriate parameters, the framework uncovers the underlying one-parameter subgroup . 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.

Paper Structure

This paper contains 45 sections, 15 theorems, 67 equations, 5 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Contributions
  3. Related work
  4. Group Equivariance
  5. Lie Groups
  6. Automatic Symmetry Discovery
  7. Background
  8. Proposed method
  9. Learning $H_\gamma$-invariant Functions: $SO(3)$
  10. Joint Discovery of $H_\gamma$ and Learning Invariant Functions
  11. One-Parameter Subgroups
  12. Learning $H_\gamma$-invariant functions: $SO(n)$
  13. Experiments
  14. Synthetic Tasks: Learning $H_\gamma$-Invariant Polynomials
  15. Real-World Symmetry Discovery
  16. ...and 30 more sections

Key Result

Theorem 4.2

Any $H_\gamma$-invariant function of the form $\Psi : X \subseteq \mathbb{R}^3 \to \mathbb{R}^m$ can be expressed as: for some $\phi : \mathbb{R}^2 \to \mathbb{R}^m$, where $\{a_i^T\}$ are rows of some $A \in SO(3)$.

Figures (5)

  • Figure 1: Block diagram of the proposed $H_\gamma^{\text{inv}}$-Net framework for discovering one-parameter subgroups of $SO(n)$ (left) and $SO(3)$ (right). The framework learns a rotation matrix $A$ and rotation rates $\lambda$ to construct a canonical invariant representation. Theorems \ref{['Thm:Hgamma-SOn']} and \ref{['Thm:Hgamma-SO3']} provide the theoretical basis for expressing any $H_\gamma$-invariant function in a standard form, facilitating the discovery of underlying one-parameter subgroups for $SO(n)$ and $SO(3)$. The model is trained end-to-end using gradient descent on $x \in [0,1]^n$ ($n$ is even for $SO(n)$ case).
  • Figure 2: Visualization of one-parameter subgroups on Lie group $G$ through the exponential map.
  • Figure 3: Visualization of the invariant representation ($\operatorname{invRep}$) and its relationship to the orbits of $H_\gamma$. Each orbit $\mathcal{O}_{H_\gamma}(x)$ is mapped to a unique invariant representation $\operatorname{invRep}(x)$, ensuring a bijective correspondence.
  • Figure 4: Matrix representation of $A_0^T A$ for the $p(x)$ task with dimension $n=3$, trained using $H_{inv}^{\gamma}$-Net, where $A_0$ denotes the ground-truth rotation matrix and $A$ is the learned rotation matrix. The matrix product highlights the alignment between the learned and true symmetry transformations.
  • Figure 5: Matrix representation of $A_0^T A$ for the $r(x)$ task with dimension $n=8$, trained using $H_{inv}^{\gamma}$-Net.Analysis shows that ground rotation matrix $A_0$ and the learnt rotation matrix $A$ have different numerical values, but they capture the one-parameter subgroups for the given functions. This relationship is confirmed by the structure of $A_0^T A$, which takes the form consisting of $2 \times 2$ blocks of scaled orthonormal matrices. The presence of this structure verifies that the learnt representation effectively captures the underlying symmetry properties, despite differences in specific numerical values.

Theorems & Definitions (37)

  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • Definition 3.6
  • Definition 3.7
  • Definition 3.8
  • Remark 4.1
  • Theorem 4.2: One-parameter subgroups of $SO(3)$
  • ...and 27 more