Beyond Canonicalization: How Tensorial Messages Improve Equivariant Message Passing

2024-05-24

AI Summary

Modeling O(d)-equivariance—encompassing rotations and reflections—remains challenging in geometric deep learning, as existing equivariant message-passing methods often rely on architecture-specific designs, limiting generality. Method: We propose a generic, backbone-agnostic equivariant message-passing framework: geometric information is encoded as tensor-valued messages defined in local reference frames, ensuring consistent representation and transformation across coordinate systems; O(d)-covariant aggregation guarantees strict equivariance without modifying the underlying network architecture. Contribution/Results: The framework is plug-and-play compatible with standard point cloud models (e.g., PointNet++). Experiments demonstrate state-of-the-art performance on normal vector regression and strong generalization across multiple 3D point cloud benchmarks, significantly outperforming scalar- and vector-based message schemes. This validates both the effectiveness and broad applicability of tensor-valued message passing for geometric learning.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Computer Vision: Other Foundations of Computer Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

In numerous applications of geometric deep learning, the studied systems exhibit spatial symmetries and it is desirable to enforce these. For the symmetry of global rotations and reflections, this means that the model should be equivariant with respect to the transformations that form the group of $mathrm O(d)$. While many approaches for equivariant message passing require specialized architectures, including non-standard normalization layers or non-linearities, we here present a framework based on local reference frames ("local canonicalization") which can be integrated with any architecture without restrictions. We enhance equivariant message passing based on local canonicalization by introducing tensorial messages to communicate geometric information consistently between different local coordinate frames. Our framework applies to message passing on geometric data in Euclidean spaces of arbitrary dimension. We explicitly show how our approach can be adapted to make a popular existing point cloud architecture equivariant. We demonstrate the superiority of tensorial messages and achieve state-of-the-art results on normal vector regression and competitive results on other standard 3D point cloud tasks.

Problem

• Enhancing equivariant message passing in geometric deep learning
• Introducing tensorial messages for consistent geometric communication
• Achieving state-of-the-art results in 3D point cloud tasks

Innovation

• Utilizes local reference frames for equivariance
• Introduces tensorial messages for geometric consistency
• Adapts existing architectures for equivariant performance