Spectral Discovery of Continuous Symmetries via Generalized Fourier Transforms

2026-03-07

AI Summary

This work proposes a novel method based on the generalized Fourier transform to discover continuous symmetries in scientific and machine learning problems, where such symmetries are often unknown and challenging to identify. By analyzing the spectral structure of functions under irreducible representations of Lie groups, the approach leverages symmetry-induced spectral sparsity to detect continuous one-parameter subgroups, circumventing conventional reliance on generator optimization or data augmentation. The method integrates representation theory, multidimensional Fourier analysis, and sparsity detection on maximal tori, offering strong theoretical grounding and enhanced interpretability. Experiments on the double pendulum system and top-quark tagging successfully recover one-dimensional continuous symmetries, demonstrating the efficacy and robustness of the proposed spectral framework.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

Continuous symmetries are fundamental to many scientific and learning problems, yet they are often unknown a priori. Existing symmetry discovery approaches typically search directly in the space of transformation generators or rely on learned augmentation schemes. We propose a fundamentally different perspective based on spectral structure. We introduce a framework for discovering continuous one-parameter subgroups using the Generalized Fourier Transform (GFT). Our central observation is that invariance to a subgroup induces structured sparsity in the spectral decomposition of a function across irreducible representations. Instead of optimizing over generators, we detect symmetries by identifying this induced sparsity pattern in the spectral domain. We develop symmetry detection procedures on maximal tori, where the GFT reduces to multi-dimensional Fourier analysis through their irreducible representations. Across structured tasks, including the double pendulum and top quark tagging, we demonstrate that spectral sparsity reliably reveals one-parameter symmetries. These results position spectral analysis as a principled and interpretable alternative to generator-based symmetry discovery.

Problem

• continuous symmetries
• symmetry discovery
• Generalized Fourier Transform
• spectral sparsity
• one-parameter subgroups

Innovation

• Generalized Fourier Transform
• continuous symmetry discovery
• spectral sparsity
• irreducible representations
• one-parameter subgroups