🤖 AI Summary
This paper addresses the challenge of reliably detecting symmetries in chaotic attractors under high noise and limited data. We propose a statistical framework integrating optimal transport and Bayesian inference: symmetry groups are selected via a Gibbs posterior constructed using the Wasserstein distance, inherently enforcing Occam’s razor, conjugate equivariance, and robustness to strong noise. Coupling Metropolis–Hastings sampling with group-action transformations enables probabilistic identification of symmetry structures. The method accurately recovers ground-truth symmetries even under severe noise and sparse observations. We validate it on human gait dynamics, revealing noise-robust, mechanically constrained symmetry evolution—uncovering how biomechanical constraints shape dynamical symmetry over time. This provides an interpretable, quantifiable paradigm for model reduction and mechanistic analysis of nonlinear dynamical systems.
📝 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.