Fast, Expressive SE$(n)$ Equivariant Networks through Weight-Sharing in Position-Orientation Space (2310.02970v3)

Abstract: Based on the theory of homogeneous spaces we derive geometrically optimal edge attributes to be used within the flexible message-passing framework. We formalize the notion of weight sharing in convolutional networks as the sharing of message functions over point-pairs that should be treated equally. We define equivalence classes of point-pairs that are identical up to a transformation in the group and derive attributes that uniquely identify these classes. Weight sharing is then obtained by conditioning message functions on these attributes. As an application of the theory, we develop an efficient equivariant group convolutional network for processing 3D point clouds. The theory of homogeneous spaces tells us how to do group convolutions with feature maps over the homogeneous space of positions $\mathbb{R}^3$, position and orientations $\mathbb{R}^3 {\times} S^2$, and the group $SE(3)$ itself. Among these, $\mathbb{R}^3 {\times} S^2$ is an optimal choice due to the ability to represent directional information, which $\mathbb{R}^3$ methods cannot, and it significantly enhances computational efficiency compared to indexing features on the full $SE(3)$ group. We support this claim with state-of-the-art results -- in accuracy and speed -- on five different benchmarks in 2D and 3D, including interatomic potential energy prediction, trajectory forecasting in N-body systems, and generating molecules via equivariant diffusion models.

• The paper introduces a novel weight-sharing approach using invariant attributes to design efficient SE(n) equivariant neural networks.
• It formalizes group convolutions in position-orientation space to reduce computational complexity while capturing isotropic and directional features.
• Empirical validations on benchmarks such as molecular generation and trajectory forecasting underscore its state-of-the-art performance.

An Analysis of SE(n) Equivariant Networks via Weight-Sharing in Position-Orientation Space

The paper "Fast, Expressive SE$(n)$ Equivariant Networks through Weight-Sharing in Position-Orientation Space" introduces a fresh approach to designing SE(n) equivariant neural networks, focusing on 3D point cloud processing. It employs the concept of weight-sharing grounded in geometric attributes, optimized for group convolutions within the position-orientation space. By formalizing weight sharing as a function of message-passing operations conditioned on invariant attributes, the paper provides a structured methodology to construct efficient and expressive group convolutional networks.

Invariant Attributes and Weight-Sharing

The authors establish an innovative framework to handle geometric data by identifying equivalence classes of point-pairs invariant to transformations in the SE(n) group. This is achieved by employing geometrically optimal edge attributes derived through the theory of homogeneous spaces. The paper thoroughly analyzes the SE(n) group action on equivalence classes and induces a parameterization involving invariant attributes. These attributes act as a means to uniquely identify equivalence classes and facilitate weight-sharing within the neural architecture.

The research articulates specific mappings for various homogeneous spaces associated with SE(n), such as position space $\mathbb{R}^n$, the hybrid position-orientation space $\mathbb{R}^3 \times S^2$, and the full group space $SE(3)$. These mappings ensure a bijective correspondence with the space of all equivalence classes, allowing a robust foundation for weight-sharing implementations across different space representations. The theoretical grounding of these mappings enhances the understanding of how invariant attributes contribute to the full expressivity and efficiency of equivariant layers.

Efficient Architecture Design

The practical application of their theoretical insights manifests in the development of a separable group convolutional framework, tailored for $\mathbb{R}^3 \times S^2$. This approach promises computational efficiency by partitioning group convolution into spatial, spherical, and channel mixing operations. By leveraging invariant attributes, the architecture reduces computational complexity without compromising on capturing intricate isotropic and directional features within point clouds.

The architecture constructs equivalence classes by implementing an effector mechanism through invariant attributes in message functions within the deep learning framework. This design choice facilitates processing equivalently under graph permutations and transformations, aligning with equivariant inductive biases intrinsic to the SE(n) group.

Empirical Validation

The paper demonstrates state-of-the-art performance across several benchmarks, notably in interatomic potential energy prediction, trajectory forecasting in N-body systems, and molecular generation tasks. The performance efficiency and accuracy underscore the potential of refining deep learning models with a strategic focus on group equivariance through geometrically aligned invariant mappings.

Implications and Future Directions

This research illuminates the prospect of enhancing neural architectures by embedding geometric and algebraic properties through invariant mappings. While its immediate application in 3D data processing is clear, the structured view of weight-sharing and convolutions furnished by invariant attributes paves the way for extending this approach to broader machine learning challenges, particularly those requiring efficiency and expressivity in high-dimensional and group-transformed data spaces.

Investigating the potential of expanding to dynamic scenarios where the group action may involve more complex transformations, or exploring other groups and homogeneous spaces, embodies plausible future research trajectories. Moreover, scaling the theoretical framework to accommodate multi-modal data and integrating with other neural network paradigms such as attention mechanisms could form innovative future explorations.

In conclusion, the paper's methodology in deriving weight-sharing schemes via invariant attributes conspicuously integrates theoretical and practical facets of deep learning, presenting a comprehensive toolkit for next-generation SE(n) equivariant networks.