Smooth, exact rotational symmetrization for deep learning on point clouds (2305.19302v3)

Abstract: Point clouds are versatile representations of 3D objects and have found widespread application in science and engineering. Many successful deep-learning models have been proposed that use them as input. The domain of chemical and materials modeling is especially challenging because exact compliance with physical constraints is highly desirable for a model to be usable in practice. These constraints include smoothness and invariance with respect to translations, rotations, and permutations of identical atoms. If these requirements are not rigorously fulfilled, atomistic simulations might lead to absurd outcomes even if the model has excellent accuracy. Consequently, dedicated architectures, which achieve invariance by restricting their design space, have been developed. General-purpose point-cloud models are more varied but often disregard rotational symmetry. We propose a general symmetrization method that adds rotational equivariance to any given model while preserving all the other requirements. Our approach simplifies the development of better atomic-scale machine-learning schemes by relaxing the constraints on the design space and making it possible to incorporate ideas that proved effective in other domains. We demonstrate this idea by introducing the Point Edge Transformer (PET) architecture, which is not intrinsically equivariant but achieves state-of-the-art performance on several benchmark datasets of molecules and solids. A-posteriori application of our general protocol makes PET exactly equivariant, with minimal changes to its accuracy.

• The paper introduces a post-training symmetrization protocol that bestows exact rotational equivariance on deep learning models for point clouds.
• The proposed approach, exemplified by the Point Edge Transformer (PET), achieves robust performance without inherent rotational symmetry.
• Empirical results on diverse datasets demonstrate that effective rotational symmetry can be attained without compromising model smoothness or accuracy.

Incompleteness of Graph Convolutional Neural Networks for Point Clouds in Three Dimensions

In the paper titled "Incompleteness of Graph Convolutional Neural Networks for Point Clouds in Three Dimensions," authors Sergey N. Pozdnyakov and Michele Ceriotti address significant constraints in applying Graph Convolutional Neural Networks (GCNNs) to point cloud representations in three-dimensional (3D) spaces. Point clouds offer a flexible and precise method of describing 3D objects, making them invaluable in various applications, including but not limited to autonomous driving, augmented reality, and materials modeling.

Context and Objective

Point clouds serve as foundational representations in the domain of geometric deep learning, where they are often engaged as inputs for complex models. A pivotal challenge in fields like chemical and materials modeling is the incorporation of physical symmetries and constraints, such as invariance to translations, rotations, and permutations. These constraints are crucial to achieving accurate and physically meaningful outputs. The authors assert that while translational and permutational invariance are generally achieved with existing architectures, rotational invariance remains a point of contention.

Methodology and Proposed Protocol

To address the challenge of rotational invariance, the authors propose a general symmetrization protocol that enhances any model by adding rotational equivariance without sacrificing other constraints like smoothness and existing invariances. This process aims to integrate disparate approaches between generic point cloud processing and models mandatory for accurate materials simulations.

The proposed protocol introduces the Point Edge Transformer (PET) architecture as an exemplar model. PET is designed not to be inherently rotationally symmetric, yet it attains state-of-the-art results across several benchmark datasets, including both molecular and solid collections. The PET model demonstrates its flexibility and enhanced capacity over previous models, while achieving post hoc equivariance through the devised protocol with minimal accuracy compromise.

Results and Implications

The paper's outcomes suggest that the explicit inclusion of equivariance within the architecture is not necessarily a prerequisite for high performance, as previously believed. Instead, adopting a symmetrization strategy post-model training allows for potentially greater architectural freedom and increased expressiveness.

The authors illustrate robust empirical results, with PET outperforming several existing solutions, employing datasets like COLL, HME21, and others. The architectural design facilitates efficient learning and computation, enhancing real-world applicability, particularly in large-scale simulations in materials science.

Future Prospects

This research opens pathways to transferring successful techniques from more general-purpose point cloud models into atom-specific applications, further bridging the divide between diverse community approaches to deep learning in geometric contexts. It encourages the community to critically evaluate the necessity of strict equivariant-functional forms within architectures, promoting exploration into advanced symmetrization techniques as a focal point for future advancement.

In conclusion, this work represents a methodical progression in the field where traditional barriers, inherent to previous models, are methodically addressed, offering a compelling argument for revisiting design paradigms in point cloud-based neural networks. This could catalyze shifts in both research focus and the deployment of AI technologies across a spectrum of domains reliant on 3D data representation.