[论文解读] Spectral Discovery of Continuous Symmetries via Generalized Fourier Transforms

Pavan Karjol, Kumar Shubham+1 more|arXiv (Cornell University)|Mar 7, 2026

Machine Learning in Materials Science被引用 0

一句话总结

本文提出基于广义傅里叶变换的框架，通过检测谱稀疏性、将输入对齐到不变量平面、并使用基于共振的正则化来恢复潜在生成元，同时学习预测映射，进而发现连续一参数对称性。

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.

研究动机与目标

• 激发在活动群未知时自动发现潜在的连续对称性
• 提出使用GFT的谱框架以识别一参数子群
• 开发具有最大环 Fourier 特征和基于共振的正则化的架构
• 在合成和真实任务上展示改进的预测和可解释的对称性恢复

提出的方法

• 通过实数范态分解 B=Q(⊕k λk J)Qᵀ 对潜在生成元 B 建模并端到端学习 Q 和 {λk}
• 通过对可学习的正交变换 Q 将输入对齐到不变量平面，以在 2D 中获得块 zk
• 在对齐坐标上构造环形傅里叶特征，使用 U(x)= {ẑm(x)} 表示边界格点中原始频率方向 m
• 施加共振正则化，通过 μ∑m (Cm⟨m,λ⟩)² 来惩罚非共振傅里叶模态以促进谱稀疏性
• 将谱特征和半径 R(x) 输入预测器 φw 映射到输出
• 训练目标为最小化标准预测损失加上共振正则化，以恢复潜在对称性参数 (Q, λ)

Figure 1: Symmetry discovery and learning framework. The input is first aligned via a learnable orthogonal transformation $Q\in SO(n)$ and decomposed into two-dimensional blocks, which are converted to polar coordinates to obtain radii and torus angles. Primitive torus Fourier features $U(x)$ , toge

实验结果

研究问题

• RQ1 谱学（GFT）视角是否能揭示潜在数据背后的单参数子群
• RQ2 如何从数据中同时学习预测映射和相应的对称性生成元
• RQ3 将输入对齐到不变量平面并执行共振是否能带来更准确的对称性恢复和更好的泛化
• RQ4 在捕捉连续旋转对称性方面，最大环傅里叶特征的有效性如何
• RQ5 与基于数据增强的对称性发现方法相比，该方法在预测性与可解释性上的表现如何

主要发现

Method	Test MSE ↓	Inv. Error ↓	Cosine Similarity ↑
Spectral Discovery	0.00298 ± 0.00024	0.00070 ± 0.00027	0.9999 ± 0.00001
Augerino	0.01053 ± 0.00040	0.00232 ± 0.00025	0.9233 ± 0.0708

• 谱学发现能够在评估任务中实现几乎与地真旋转对称性对齐的完美度（余弦相似度约 ≈ 1.0）
• 在双摆任务中，谱学发现的测试均方误差和不变量误差显著低于 Augerino，余弦相似度约 ≈ 0.9999
• 在顶夸克标记任务中，谱学发现提升了准确率并实现接近完美的生成元恢复（余弦相似度约 ≈ 0.9999）
• 基于共振的正则化与环形傅里叶特征实现对潜在单参数对称性的可解释且稳定的恢复，同时保持竞争性的预测性能

Figure 2: Recovered vs. True Rotational Generators. Comparison of the learned (left) and ground-truth (right) generators for (a) the Double Pendulum system (diagonal $\Delta(SO(2))$ generator) and (b) the Top Tagging task. Both learned generators demonstrate near-perfect alignment with the physical