Title: Adaptive Canonicalization with Application to Invariant Anisotropic Geometric Networks
Primary area: learning on graphs and other geometries & topologies
Keywords: Equivariant machine learning, Canonicalization, Universal approximation, Classification, Graph neural networks, Spectral methods, Point cloud networks, Anisotropic networks

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 *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.

TL;DR: We present adaptive canonicalization, a symmetry-preserving and continuous method with universal approximation properties for equivariant machine learning.