何时以及如何做规范化：一项泛化视角的理论分析

View PDF HTML (experimental)

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)