aiqbench analysis
This paper provides a rigorous theoretical framework for canonization in non-invariant backbones, establishing that the generalization error depends critically on the regularity of the chosen method. By proving that Hilbert curve serialization offers polynomial growth in covering numbers compared to the exponential growth of lexicographical sorting, it formally validates its empirical success in point cloud processing architectures. The work bridges the gap between heuristic data augmentation strategies and formal learning theory, offering clear guidelines for selecting optimal canonization techniques.
When and How to Canonize: A Generalization Perspective
arXiv:2605.11008v2 Announce Type: replace 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 https://github.com/yonatansverdlov/Canonization
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
Machine Learning (cs.LG); Artificial Intelligence (cs.AI)
arXiv:2605.11008 [cs.LG]
(or arXiv:2605.11008v2 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2605.11008 Focus to learn more arXiv-issued DOI via DataCite
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Full text under the publisher's open licence · 289 words
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