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A Bayesian Framework for Symmetry Inference in Chaotic Attractors

Ziad Ghanem, Chang Hyunwoong, Preskella Mrad

TL;DR

This work tackles the problem of identifying symmetry structures in chaotic attractors from partial, noisy observations by casting symmetry inference as Bayesian model selection over a lattice of subgroup hypotheses. It introduces a group-orbit Gibbs posterior built from the average $d(X, \sigma X)^2$ with $d = W_2$, providing a principled, uncertainty-aware alternative to threshold-based symmetry tests. The main theoretical contributions are a Bayesian Occam's razor that favors the simplest consistent symmetry, conjugation equivariance ensuring frame-independence, and stability bounds under data perturbations for robustness and uncertainty quantification; posterior inference is carried out with Metropolis–Hastings in a discrete subgroup space. Empirically, the framework accurately recovers symmetries in noisy synthetic and equivariant systems, handles nested dihedral structures with multimodal posteriors, and reveals hierarchy‑shifting symmetry patterns in real-world human gait data under mechanical constraints, highlighting its practical utility for dynamical system analysis and biomechanics.

Abstract

Detecting symmetry from data is a fundamental problem in signal analysis, providing insight into underlying structure and constraints. When data emerge as trajectories of dynamical systems, symmetries encode structural properties of the dynamics that enable model reduction, principled comparison across conditions, and detection of regime changes. While recent optimal transport methods provide practical tools for data-driven symmetry detection in this setting, they rely on deterministic thresholds and lack uncertainty quantification, limiting robustness to noise and ability to resolve hierarchical symmetry structures. We present a Bayesian framework that formulates symmetry detection as probabilistic model selection over a lattice of candidate subgroups, using a Gibbs posterior constructed from Wasserstein distances between observed data and group-transformed copies. We establish three theoretical guarantees: $(i)$ a Bayesian Occam's razor favoring minimal symmetry consistent with data, $(ii)$ conjugation equivariance ensuring frame-independence, and $(iii)$ stability bounds under perturbations for robustness to noise. Posterior inference is performed via Metropolis-Hastings sampling and numerical experiments on equivariant dynamical systems and synthetic point clouds demonstrate accurate symmetry recovery under high noise and small sample sizes. An application to human gait dynamics reveals symmetry changes induced by mechanical constraints, demonstrating the framework's utility for statistical inference in biomechanical and dynamical systems.

A Bayesian Framework for Symmetry Inference in Chaotic Attractors

TL;DR

This work tackles the problem of identifying symmetry structures in chaotic attractors from partial, noisy observations by casting symmetry inference as Bayesian model selection over a lattice of subgroup hypotheses. It introduces a group-orbit Gibbs posterior built from the average with , providing a principled, uncertainty-aware alternative to threshold-based symmetry tests. The main theoretical contributions are a Bayesian Occam's razor that favors the simplest consistent symmetry, conjugation equivariance ensuring frame-independence, and stability bounds under data perturbations for robustness and uncertainty quantification; posterior inference is carried out with Metropolis–Hastings in a discrete subgroup space. Empirically, the framework accurately recovers symmetries in noisy synthetic and equivariant systems, handles nested dihedral structures with multimodal posteriors, and reveals hierarchy‑shifting symmetry patterns in real-world human gait data under mechanical constraints, highlighting its practical utility for dynamical system analysis and biomechanics.

Abstract

Detecting symmetry from data is a fundamental problem in signal analysis, providing insight into underlying structure and constraints. When data emerge as trajectories of dynamical systems, symmetries encode structural properties of the dynamics that enable model reduction, principled comparison across conditions, and detection of regime changes. While recent optimal transport methods provide practical tools for data-driven symmetry detection in this setting, they rely on deterministic thresholds and lack uncertainty quantification, limiting robustness to noise and ability to resolve hierarchical symmetry structures. We present a Bayesian framework that formulates symmetry detection as probabilistic model selection over a lattice of candidate subgroups, using a Gibbs posterior constructed from Wasserstein distances between observed data and group-transformed copies. We establish three theoretical guarantees: a Bayesian Occam's razor favoring minimal symmetry consistent with data, conjugation equivariance ensuring frame-independence, and stability bounds under perturbations for robustness to noise. Posterior inference is performed via Metropolis-Hastings sampling and numerical experiments on equivariant dynamical systems and synthetic point clouds demonstrate accurate symmetry recovery under high noise and small sample sizes. An application to human gait dynamics reveals symmetry changes induced by mechanical constraints, demonstrating the framework's utility for statistical inference in biomechanical and dynamical systems.
Paper Structure (25 sections, 3 theorems, 35 equations, 10 figures)

This paper contains 25 sections, 3 theorems, 35 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Methodology
  3. Preliminaries: The Structure of Symmetric Attractors
  4. The Wasserstein Distance as a Symmetry Metric
  5. A Bayesian Framework for Symmetry Inference
  6. The Cost Function and its Properties
  7. Posterior Distribution
  8. The MCMC Sampler
  9. Numerical Validation
  10. Example 1: Dihedral Symmetries in The Chossat-Golubitsky System
  11. A Deterministic Benchmark
  12. Bayesian Inference Results
  13. Example 2: Nested Dihedral Symmetries in Synthetic Point Cloud
  14. A Deterministic Benchmark
  15. Bayesian Inference Results
  16. ...and 10 more sections

Key Result

Proposition 3.1

Let $\Sigma_S, \Sigma_L$ be two candidate symmetry groups for $X$ with $\Sigma_S < \Sigma_L$. Then, one has if and only if the average cost of the transformations of $X$ contained in $\Sigma_S$ is less than the average cost of the transformations in $\Sigma_L \setminus \Sigma_S$.

Figures (10)

  • Figure 1: (Left) Noise-free attractor for the Chossat--Golubitsky map, generated from a $50,000$-point trajectory to illustrate the ground-truth $D_3$ symmetry. (Right) Noisy ($\sigma = 0.5$) $150$ point trajectory used for analysis, where the symmetry is visually obscured.
  • Figure 2: Benchmark analysis of the deterministic method from Cisternas2025 on the noisy $D_3$ attractor. (Left) Distribution of squared Wasserstein distances for candidate groups; distances for $D_3$ cluster near zero. (Right) Classification outcome as a function of threshold $\upsilon$. Correct identification of $n = 3$ occurs only within the narrow green "robust window".
  • Figure 3: Bayesian inference on the noisy $D_3$ attractor. (Left) MCMC trace for symmetry order $n$. (Right) Posterior distribution $P(n \mid X)$, with MAP at $n = 3$ (posterior probability $72.8\%$).
  • Figure 4: (Left) Noise-free point cloud with perfect $D_{12}$ symmetry. (Right) Noisy dataset ($N = 192$, $\sigma = 0.05$) used for analysis.
  • Figure 5: Benchmark analysis from Cisternas2025 on the noisy $D_{12}$ dataset. (Left) Boxplot of squared Wasserstein distances for candidate groups; all subgroups of $D_{12}$ (highlighted in green) exhibit low costs. (Right) Threshold sweep fails to isolate the true symmetry $n = 12$.
  • ...and 5 more figures

Theorems & Definitions (7)

  • Proposition 3.1: Occam's Razor
  • proof
  • Proposition 3.2: Conjugation Equivariance of Cost
  • proof
  • Proposition 3.3: Stability to Perturbations
  • proof
  • Remark A.1