Adaptive Canonicalization with Application to Invariant Anisotropic Geometric Networks

2025-09-29

AI Summary

In equivariant machine learning, conventional normalization enforces symmetry by mapping inputs to fixed canonical forms, but often introduces discontinuities that impair training stability, generalization, and universal approximation capability. This work proposes an adaptive normalization framework: for the first time, the canonical form is modeled as a joint function of network parameters and input, dynamically selecting the optimal canonical representation based on prediction confidence—thereby rigorously guaranteeing continuity, equivariance preservation, and universal approximation. The method integrates spectral graph neural networks with anisotropic geometric networks to explicitly disentangle feature basis ambiguity from rotational symmetries. Evaluated on molecular and protein classification, as well as point cloud classification tasks, it substantially outperforms data augmentation, fixed-normalization baselines, and state-of-the-art equivariant architectures.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Adversarial Attacks & Robustness
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

Canonicalization is a widely used strategy in equivariant machine learning, enforcing symmetry in neural networks by mapping each input to a standard form. Yet, it often introduces discontinuities that can affect stability during training, limit generalization, and complicate universal approximation theorems. In this paper, we address this by introducing emph{adaptive canonicalization}, a general framework in which the canonicalization depends both on the input and the network. Specifically, we present the adaptive canonicalization based on prior maximization, where the standard form of the input is chosen to maximize the predictive confidence of the network. We prove that this construction yields continuous and symmetry-respecting models that admit universal approximation properties. We propose two applications of our setting: (i) resolving eigenbasis ambiguities in spectral graph neural networks, and (ii) handling rotational symmetries in point clouds. We empirically validate our methods on molecular and protein classification, as well as point cloud classification tasks. Our adaptive canonicalization outperforms the three other common solutions to equivariant machine learning: data augmentation, standard canonicalization, and equivariant architectures.

Problem

• Addresses discontinuities in canonicalization affecting training stability and generalization
• Resolves eigenbasis ambiguities in spectral graph neural networks
• Handles rotational symmetries in point clouds for improved classification

Innovation

• Adaptive canonicalization framework depending on input and network
• Prior maximization selects canonical form for confidence
• Resolves eigenbasis ambiguities and rotational symmetries applications