🤖 AI Summary
This work addresses the lack of a systematic theoretical understanding of the generalization performance of invariance-inducing methods such as canonicalization and group averaging. The authors develop a unified theoretical framework to analyze the generalization error of canonicalization, group averaging, and structurally invariant models through upper bounds on covering numbers, establishing a hierarchy among their generalization capabilities. They provide the first rigorous proof that the effectiveness of canonicalization critically depends on its regularity properties, revealing that Hilbert curve-based canonicalization outperforms lexicographic ordering in point cloud tasks because it ensures polynomial growth of covering numbers, whereas lexicographic ordering leads to exponential growth. This theory underpins state-of-the-art point cloud architectures and is corroborated by empirical validation.
📝 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