“Fast, Expressive SE(n) Equivariant Networks through WeightSharing in PositionOrientation Space”

ABSTRACT
Based on the theory of homogeneous spaces we derive \textit{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 pointpairs that should be treated equally. We define equivalence classes of pointpairs 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 ℝ3, position and orientations ℝ3×S2, and the group SE(3) itself. Among these, ℝ3×S2 is an optimal choice due to the ability to represent directional information, which ℝ3 methods cannot, and it significantly enhances computational efficiency compared to indexing features on the full SE(3) group. We empirically support this claim by reaching stateoftheart results — in accuracy and speed — on three different benchmarks: interatomic potential energy prediction, trajectory forecasting in Nbody systems, and generating molecules via equivariant diffusion models.

PRESENTER
Erik Bekkers is an assistant professor in Geometric Deep Learning in the Machine Learning Lab of the University of Amsterdam (AMLab, UvA). Before this he did a postdoc in applied differential geometry at the dept. of Applied Mathematics at Technical University Eindhoven (TU/e). In his PhD (cum laude, Biomedical Engineering, TU/e), he developed medical image analysis algorithms based on subRiemannian geometry in the Lie group SE(2) using the same mathematical principles that underlie mathematical models of human visual perception. Such mathematics find their application in machine learning where through symmetries and geometric structure, robust and efficient representation learning methods are obtained. His current work is on generalizations of group convolutional NNs and improvements of computational and representation efficiency through sparse, adaptive, and geometric learning mechanisms. Erik is a recipient of a MICCAI Young Scientist Award 2018, Philips Impact Award (MIDL 2018) and a VENI personal research grant (awarded by the Dutch Research Council (NWO)). Erik is recognized as an ELLIS Scholar in the ELLIS program on Geometric Deep Learning.