Interpretable Discovery of One-parameter Subgroups: A Modular Framework for Elliptical, Hyperbolic, and Parabolic Symmetries

We propose a modular, data-driven framework for jointly learning unknown functional mappings and discovering the underlying one-parameter symmetry subgroup governing the data. Unlike conventional geometric deep learning methods that assume known symmetries, our approach identifies the relevant continuous subgroup directly from data. We consider the broad class of one-parameter subgroups, which admit a canonical geometric classification into three regimes: elliptical, hyperbolic, and parabolic. Given an assumed regime, our framework instantiates a corresponding symmetry discovery architecture with invariant and equivariant representation layers structured according to the Lie algebra of the subgroup, and learns the exact generator parameters end-to-end from data. This yields models whose invariance or equivariance is guaranteed by construction and admits formal proofs, enabling symmetry to be explicitly traced to identifiable components of the architecture. The approach is applicable to one-parameter subgroups of a wide range of matrix Lie groups, including $SO(n)$, $SL(n)$, and the Lorentz group. Experiments on synthetic and real-world systems, including moment of inertia prediction, double-pendulum dynamics, and high-energy \textit{Top Quark Tagging}, demonstrate accurate subgroup recovery and strong predictive performance across both compact and non-compact regimes.

💡 Research Summary

The paper introduces a novel, modular framework that simultaneously learns an unknown target function and discovers the underlying one‑parameter symmetry subgroup governing the data. Traditional geometric deep learning methods embed equivariance or invariance by hard‑coding a known symmetry group (e.g., SO(3), SE(3)) into the architecture. However, many scientific and engineering problems involve continuous symmetries that are not known a priori, and imposing an incorrect group can degrade performance. To address this gap, the authors focus on one‑parameter subgroups—connected 1‑dimensional Lie subgroups generated by the exponential of a single Lie algebra element. Such subgroups naturally fall into three geometric regimes: elliptic (compact, rotation‑like), hyperbolic (non‑compact, squeeze/boost‑like), and parabolic (non‑compact, shear‑like).

The core contribution is a regime‑specific architecture that enforces invariance (or equivariance) by construction. For a chosen regime, the input vector (x\in\mathbb{R}^n) is first linearly transformed by a learnable matrix (A) into a coordinate system aligned with the unknown subgroup. The transformed vector is partitioned into 2‑dimensional blocks (v_i). The first block (v_1) is used to solve for a canonical group parameter (t_0) (e.g., a rotation angle, a boost rapidity, or a shear amount) that aligns (v_1) with a fixed reference configuration. The same transformation is applied uniformly to all blocks, effectively mapping the entire input onto a canonical orbit representative (z = \text{invRep}(x)). This mapping collapses each orbit of the unknown subgroup to a single point, guaranteeing invariance while preserving all information needed for the downstream task.

The learnable parameters of the model consist of: (1) the orientation matrix (A), (2) the set of regime‑specific generator coefficients ({\lambda_k}) that define the canonical Lie algebra element (\hat B), and (3) the parameters of a downstream predictor (\phi) that operates on the invariant representation (z). Training proceeds end‑to‑end by minimizing a task‑specific loss (e.g., regression or classification) together with optional regularizers on the generator parameters. Because the invariance is built into the architecture, the loss landscape directly informs the discovery of the correct subgroup; the learned (\lambda_k) and (A) can be inspected to retrieve the exact symmetry transformation.

Mathematically, the paper provides rigorous proofs that (i) the invariant representation is orbit‑separating (different orbits map to distinct (z)), and (ii) the overall network is provably invariant under the discovered subgroup. These results hold for any matrix Lie group that admits a one‑parameter subgroup, including (SO(n)), (SL(n)), and the Lorentz group, making the approach broadly applicable.

Empirical validation spans synthetic experiments—where random generators and orientations are sampled and the model successfully recovers them—to real‑world domains:

• Moment of inertia prediction: The data exhibits rotational symmetry; the model automatically discovers an elliptic subgroup and matches or exceeds the performance of handcrafted rotation‑equivariant networks.
• Double‑pendulum dynamics: The underlying physics possesses a hyperbolic (boost‑like) symmetry in phase space; the framework learns the corresponding generator, leading to more data‑efficient learning of the dynamics.
• High‑energy top‑quark tagging: Jet images contain non‑compact, shear‑type symmetries; the parabolic regime of the model captures these, achieving higher tagging accuracy than baselines such as Augerino, LieGAN, and Bispectral Neural Networks.

A comparative table highlights that, unlike prior symmetry‑discovery methods (Augerino, LieGG, LieGAN, etc.), the proposed Hγ‑Net integrates discovery and prediction in a single joint training loop, provides an interpretable predictor (the invariant layers are explicitly tied to the learned subgroup), and supports the full spectrum of 1‑D continuous groups rather than being limited to compact or discretized approximations.

In conclusion, the paper delivers a unified, mathematically grounded framework for discovering and exploiting one‑parameter symmetries directly from data. By modularizing the architecture into regime‑specific invariant layers and a downstream predictor, it offers both strong empirical performance and transparent interpretability—key desiderata for scientific machine learning. Future directions include extending the method to multi‑parameter subgroups, automating regime selection, and applying the approach to more complex physical systems where hidden continuous symmetries play a crucial role.