Vivek V Kashyap | Searcharxiv

arXiv2026

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.

cs.LG↗

arXiv2025

Learning Equivariant Functions via Quadratic Forms

In this study, we introduce a method for learning group (known or unknown) equivariant functions by learning the associated quadratic form $x^T A x$ corresponding to the group from the data. Certain groups, known as orthogonal groups, preserve a specific quadratic form, and we leverage this property to uncover the underlying symmetry group under the assumption that it is orthogonal. By utilizing the corresponding unique symmetric matrix and its inherent diagonal form, we incorporate suitable inductive biases into the neural network architecture, leading to models that are both simplified and efficient. Our approach results in an invariant model that preserves norms, while the equivariant model is represented as a product of a norm-invariant model and a scale-invariant model, where the ``product'' refers to the group action. Moreover, we extend our framework to a more general setting where the function acts on tuples of input vectors via a diagonal (or product) group action. In this extension, the equivariant function is decomposed into an angular component extracted solely from the normalized first vector and a scale-invariant component that depends on the full Gram matrix of the tuple. This decomposition captures the inter-dependencies between multiple inputs while preserving the underlying group symmetry. We assess the effectiveness of our framework across multiple tasks, including polynomial regression, top quark tagging, and moment of inertia matrix prediction. Comparative analysis with baseline methods demonstrates that our model consistently excels in both discovering the underlying symmetry and efficiently learning the corresponding equivariant function.

cs.LG↗