Symmetry-to-Symmetry Learning Dynamics

• The paper's main contribution is the formal integration of Lie groups and Lie derivatives to enforce model equivariance and reveal intrinsic symmetries.
• It demonstrates the use of convex regularization to promote symmetry in learning, leading to enhanced robustness, reduced overfitting, and improved generalization.
• The work outlines algorithmic implementations across various architectures, offering practical strategies for symmetry-driven efficiency and sample complexity reduction.

Symmetry-to-Symmetry Learning Dynamics encompasses theoretical and algorithmic principles in which the identification, imposition, preservation, or discovery of symmetry structures fundamentally shapes the behavior, efficiency, and generalization capability of learning systems. In this domain, symmetry refers to invariance or equivariance of functions (such as neural networks, dynamical system models, or loss functions) under the action of transformation groups. Learning dynamics are then directly constrained or influenced by these group actions, both at the parameter and functional levels, yielding conservation laws, reduced sample complexity, structural regularization, and deep insight into phase transitions and overparameterization in modern machine learning.

1. Mathematical Frameworks for Symmetry Learning

Central to modern symmetry learning is the explicit mathematical representation of symmetries via group actions—often Lie groups—and their associated Lie algebras. Machine learning models are typically cast as smooth maps $F: V \to W$ or as sections of vector bundles $\pi: E \to M$. If a group $G$ acts on these spaces, $F$ is said to be equivariant if: $$K_g F = F \quad \forall g \in G, \quad K_g F (p) = \Theta_{g^{-1}}\bigl(F(\theta(p, g))\bigr)$$ The infinitesimal generator of this action—the Lie derivative—plays a central role: $$\mathcal{L}_\xi F = \left.\frac{d}{dt}\right|_{t=0} K_{\exp(t\xi)} F$$ where $\xi$ is an element of the Lie algebra $\mathfrak{g}$. Setting $\mathcal{L}_\xi F = 0$ enforces equivariance; conversely, discovering which $\xi$ satisfy this for a given model identifies its symmetries (Otto et al., 2023).

Within dynamical systems, symmetry is reflected in the invariance of the governing vector field $F$ under group actions on state and control, e.g.: $$F(\phi_g(x), \psi_g(u)) = \phi_g(F(x, u)) \quad \forall g \in G$$ where $\phi_g, \psi_g$ define the action on state and input, respectively (Sonmez et al., 27 Mar 2024).

2. Duality of Symmetry Imposition and Discovery

The Lie derivative structure leads to a duality:

• Enforcing symmetry involves constraining models to the nullspace of the Lie derivative ($\mathcal{L}_\xi F = 0$), ensuring all relevant symmetry directions act trivially;
• Discovering symmetries leverages the linear operator $L_F: \xi \mapsto \mathcal{L}_\xi F$, where the nullspace $\operatorname{Null}(L_F)$ identifies the Lie algebra of all symmetries possessed by $F$.

Bilinearity of the Lie derivative underpins this duality, and both aspects can be implemented via efficient linear–algebraic methods (Otto et al., 2023). Theoretical results guarantee that the largest connected symmetry subgroup is characterized by this nullspace, enabling symmetry exploitation in broad function spaces and model classes.

3. Convex Regularization and Symmetry Promotion

To promote but not strictly enforce symmetry—especially when data are only approximately symmetric—convex regularizers are built from the singular spectrum of the symmetry operator: $$R_{G, *}(F) = \|L_F\|_* = \sum_{i=1}^{\dim G} \sigma_i(L_F)$$ where $\sigma_i$ are singular values. Penalizing $R_{G, *}(F)$ ensures the model remains as symmetric as the data allows, biasing the solution toward maximal symmetry consistent with empirical evidence. For discrete groups $G$, the analogous penalty is a group sum: $R_{G, 1}(F) = \sum_{g \in G} \| K_g F - F \|$ These regularizations yield models with greater robustness, increased data efficiency, and improved extrapolation, as symmetry acts effectively as a structural prior (Otto et al., 2023).

4. Algorithmic Implementations Across Model Classes

The unified framework is applicable to diverse model architectures:

Model Class	Symmetry Integration Mechanism	Representative Equation/Formulation
Basis Function Regression	Constrain regression weights or regularize via nuclear norm	$$\min_W \sum_j \|y_j - W D(x_j)\|^2 + \gamma R_{G,*}(W D)$$
Dynamical Systems Discovery	Equivariance enforced on vector fields by Lie derivative	$$\mathcal{L}_\xi F(x) = \phi(\xi)F(x) - \frac{\partial F}{\partial x} \phi(\xi)x = 0$$
Neural Networks	Weight tying and bias constraints propagate equivariance layerwise	$$\phi_l(\xi) W^{(l)} = W^{(l)} \phi_{l-1}(\xi),\ \phi_l(\xi) b^{(l)} = 0$$
Neural Operators	Group-constrained kernel constructions, derivative vanishing	See detailed expressions for kernels and their Lie derivatives (Otto et al., 2023)

In all classes, enforcing symmetry reduces effective model capacity, leading to sparser, lower-rank, or more parsimonious solutions (Ziyin, 2023). Models trained with symmetry-promoting regularizers have been shown to generalize better in domains as varied as image classification (translation invariance), molecular dynamics (rotational invariance), and the discovery of physical conservation laws (Otto et al., 2023, Wang et al., 2020).

5. The Role of Lie Groups, Locality, and Representation Spaces

In the most general setting, the group $G$ is a Lie group acting fiber-linearly. Its algebraic structure facilitates the construction of meaningful invariants and equivariants at all levels of function representations, including dynamical operators, integral kernels, graph neural network structures, and physical modeling frameworks.

For dynamical models, Cartan’s moving frame and invariantization methods (associated to coset space decompositions and cross-sections of group action) are used to factor out symmetry-related redundancies, yielding canonical coordinate systems for learning and prediction (Sonmez et al., 27 Mar 2024). These support sample-efficient learning, particularly in reinforcement learning and control when only the dynamics (but not the reward) possess the symmetry.

6. Theoretical Guarantees and Practical Implications

Formal results show:

• The nullspace of the Lie derivative characterizes all intrinsic connected symmetries;
• Nuclear norm regularization preferentially selects high-symmetry solutions (connected to compressed sensing and robust PCA guarantees on low-rank recovery);
• Equivariant representations allow for data efficient learning and improved extrapolation, as one example of symmetry-to-symmetry dynamics;
• Under weight decay or strong noise, solutions converge onto constrained subspaces (e.g., O-mirror planes), yielding sparsity, low rankness, or homogeneous ensembling depending on the type of symmetry present (Ziyin, 2023).

Practically, these results translate to improved extrapolation, greater robustness under distribution shift, and the ability to incorporate known invariances of physical systems directly into learned models—often dramatically reducing training requirements and mitigating overfitting (Bergman, 2018, Wang et al., 2020, Lee et al., 2022).

7. Open Problems and Future Directions

Several challenges and research avenues remain:

• Selection or estimation of candidate symmetry groups $G$, especially in settings where the "correct" symmetry is not obvious;
• Extending the framework to treat approximate, partially broken, or non-continuous symmetries;
• Generalizing to jet bundles for partial differential equations, integral operators on fields, or complex multi-body systems;
• Optimization algorithm development for efficient nuclear norm minimization in high-dimensional operator spaces;
• Understanding sensitivities to discretization, basis selection, and group representation choices.

Future work is also anticipated to connect symmetry-based frameworks with Koopman operator theory, more deeply integrate uncertainty quantification (e.g., Bayesian methods), and explore “geometric” regularizations in deep learning, including adaptation to time-varying or data-dependent symmetries.

Summary

Symmetry-to-symmetry learning dynamics encapsulate the use of group-theoretic principles—expressed via linear operators such as the Lie derivative—to systematically enforce, discover, and promote symmetry in machine learning models. This approach unifies architectural, algorithmic, and statistical perspectives, yielding a framework in which symmetry is central: reducing sample complexity, imposing interpretable constraints, guiding gradient flow, and supporting robustness and generalization. By leveraging symmetry-aware regularization and representation, the field moves toward mathematically rigorous, data-efficient, and physically meaningful learning paradigms that are naturally extensible across domains—while also motivating rich new directions in theory and practice (Otto et al., 2023).