A Bayesian framework for symmetry inference in chaotic attractors.

Area of Science:

• Dynamical Systems and Signal Analysis
• Statistical Inference and Machine Learning
• Computational Mathematics

Background:

• Symmetry detection is crucial for understanding data structure in signal analysis and dynamical systems.
• Existing optimal transport methods for symmetry detection lack uncertainty quantification and robustness to noise.
• Hierarchical symmetry structures are challenging to resolve with current deterministic approaches.

Purpose of the Study:

• To develop a Bayesian framework for robust and uncertainty-aware symmetry detection in dynamical systems.
• To formulate symmetry detection as probabilistic model selection over a lattice of candidate subgroups.
• To provide theoretical guarantees for accuracy, frame-independence, and noise robustness.

Main Methods:

• A Bayesian framework using a Gibbs posterior based on Wasserstein distances.
• Probabilistic model selection over a lattice of candidate symmetry subgroups.
• Metropolis-Hastings sampling for posterior inference.

Main Results:

• Theoretical guarantees including Bayesian Occam's razor, conjugation equivariance, and stability bounds.
• Accurate symmetry recovery in numerical experiments with equivariant dynamical systems and point clouds, even under high noise and small sample sizes.
• Demonstrated utility in analyzing human gait dynamics, revealing symmetry degradation due to mechanical constraints.

Conclusions:

• The proposed Bayesian framework offers a principled and robust approach to symmetry detection in complex data.
• The method provides uncertainty quantification and handles noise effectively, outperforming existing techniques.
• Applications in biomechanics and dynamical systems highlight its practical value for statistical inference.