Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Continuous Symmetry Discovery and Enforcement Using Infinitesimal Generators of Multi-parameter Group Actions

Ben Shaw, Sasidhar Kunapuli, Abram Magner, Kevin R. Moon

TL;DR

This work addresses discovering continuous symmetries beyond affine transformations by estimating infinitesimal generators of multi-parameter group actions and automatically inferring the number of independent generators. It introduces isometry-focused restrictions via Killing vectors to constrain the search space and develops symmetry-enforcement methods based on invariant-feature construction and an explicit regularization term, with a robustness analysis. The approach is demonstrated on neural networks, non-affine symmetries, and infinitesimal isometries, showing improved generalization and reduced need for data augmentation. The results suggest a practical path toward leveraging symmetry in complex ML tasks with non-affine and metric-aware transformations.

Abstract

Symmetry-informed machine learning can exhibit advantages over machine learning which fails to account for symmetry. In the context of continuous symmetry detection, current state of the art experiments are largely limited to detecting affine transformations. Herein, we outline a computationally efficient framework for discovering infinitesimal generators of multi-parameter group actions which are not generally affine transformations. This framework accommodates the automatic discovery of the number of linearly independent infinitesimal generators. We build upon recent work in continuous symmetry discovery by extending to neural networks and by restricting the symmetry search space to infinitesimal isometries. We also introduce symmetry enforcement of smooth models using vector field regularization, thereby improving model generalization. The notion of vector field similarity is also generalized for non-Euclidean Riemannian metric tensors.

Continuous Symmetry Discovery and Enforcement Using Infinitesimal Generators of Multi-parameter Group Actions

TL;DR

This work addresses discovering continuous symmetries beyond affine transformations by estimating infinitesimal generators of multi-parameter group actions and automatically inferring the number of independent generators. It introduces isometry-focused restrictions via Killing vectors to constrain the search space and develops symmetry-enforcement methods based on invariant-feature construction and an explicit regularization term, with a robustness analysis. The approach is demonstrated on neural networks, non-affine symmetries, and infinitesimal isometries, showing improved generalization and reduced need for data augmentation. The results suggest a practical path toward leveraging symmetry in complex ML tasks with non-affine and metric-aware transformations.

Abstract

Symmetry-informed machine learning can exhibit advantages over machine learning which fails to account for symmetry. In the context of continuous symmetry detection, current state of the art experiments are largely limited to detecting affine transformations. Herein, we outline a computationally efficient framework for discovering infinitesimal generators of multi-parameter group actions which are not generally affine transformations. This framework accommodates the automatic discovery of the number of linearly independent infinitesimal generators. We build upon recent work in continuous symmetry discovery by extending to neural networks and by restricting the symmetry search space to infinitesimal isometries. We also introduce symmetry enforcement of smooth models using vector field regularization, thereby improving model generalization. The notion of vector field similarity is also generalized for non-Euclidean Riemannian metric tensors.
Paper Structure (28 sections, 4 theorems, 30 equations, 1 figure, 1 algorithm)

This paper contains 28 sections, 4 theorems, 30 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Symmetry Discovery
  4. Symmetry Enforcement
  5. Background
  6. Vector Fields and Flows
  7. Vector Fields and Symmetry
  8. Infinitesimal Generators of Multi-Parameter Group Actions
  9. Killing Vectors and Isometries
  10. Problem Formulation
  11. Methods
  12. Estimating the Infinitesimal Generators
  13. Isometry Discovery
  14. A Note about the Freedom Present in Identifying Vector Field Symmetries
  15. Robustness analysis
  16. ...and 13 more sections

Key Result

Theorem 5.1

Let $f:\mathbb{R}^n \to \mathbb{R}$ and $\hat{f}:\mathbb{R}^n \to \mathbb{R}$ be $C_{\infty}$ functions. Suppose that $\|f - \hat{f}\|_{\infty} \leq \epsilon$. Suppose that $\hat{X}$ is a vector field satisfying $\hat{X}(\hat{f}) = 0$ and satisfies the following Lipschitz property: for any smooth fu Let $\hat{\Psi}(t, z)$ be the flow associated with $\hat{X}$. Then As a consequence, we have, for

Figures (1)

  • Figure 1: Symmetry enforcement visualizations for the second experiment. Top left: an illustration of the training data and regression labels. Top right: a contour plot of the ground truth regression function. Bottom left: a contour plot of the model trained without symmetry enforcement. Bottom right: a contour plot of a model trained by enforcing symmetry.

Theorems & Definitions (10)

  • Theorem 5.1
  • Proof 1
  • Definition 5.2: Completeness and soundness
  • Definition 5.3: Approximation target class
  • Lemma 5.4: Small $Xg$ in $\hat{H}_k$ implies closeness to $\text{span}\{f_1, ..., f_k\}$
  • Proof 2
  • Theorem 5.5: Approximating the approximation target class via approximate kernels
  • Proof 3
  • Theorem 5.6: Soundness
  • Proof 4