arXiv cs.AI· Yonatan Sverdlov, Benjamin Friedman, Snir Hordan, Nadav Dym·· 3 天前AI 评分22
何时以及如何做规范化：一个泛化视角
When and How to Canonize: A Generalization Perspective
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研究提出理论框架，通过覆盖数界定群平均与规范化方法在非不变骨干网络上的泛化误差，证明规范化模型的误差界最好与结构不变和群平均模型持平，最差与非不变基线持平，且取决于规范化的正则性。在点云置换群上，该框架证明字典序排序的覆盖数随维度指数增长，而 Hilbert 曲线规范化保证多项式增长，首次为 Hilbert 曲线序列化在 SOTA 点云架构中的成功提供形式化理论依据。
正文 · 原文
Abstract:While invariant architectures are standard for processing symmetric data, there is growing interest in achieving invariance by applying group averaging or canonization to non-invariant backbones. However, the theoretical generalization properties of these alternative strategies remain poorly understood. We introduce a theoretical framework to analyze the generalization error of these methods by bounding their covering numbers. We establish a rigorous generalization hierarchy: the error bounds of canonized models are at best equal to the error bounds of structurally invariant and group-averaged models, and at worst equal to the bounds of non-invariant baselines. Furthermore, we show that there exist optimal canonizations which attain the optimal error bounds, and poor canonizations which attain the non-invariant error bounds, and that this depends on the regularity of the canonization. Finally, applying this framework to permutation groups in point cloud processing, we rigorously prove that the covering number of lexicographical sorting grows exponentially with point cloud dimension, whereas Hilbert curve canonization guarantees polynomial growth. This provides the first formal theoretical justification for the empirical success of Hilbert curve serialization in state-of-the-art point cloud architectures. We conclude with experiments that support our theoretical claims.
Code is available at this https URL
| Subjects: | Machine Learning (cs.LG); Artificial Intelligence (cs.AI) |
| Cite as: | arXiv:2605.11008 [cs.LG] |
| (or arXiv:2605.11008v3 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2605.11008 arXiv-issued DOI via DataCite |
Submission history
From: Yonatan Sverdlov [view email]
[v1]
Sun, 10 May 2026 08:51:37 UTC (187 KB)
[v2]
Sun, 27 Sep 2026 09:47:44 UTC (188 KB)
[v3]
Sun, 11 Oct 2026 13:50:43 UTC (188 KB)
来源：arXiv cs.AI · arxiv.org