Geometric Deep Learning

Many problems in chemistry and biology involve data with geometric structure: molecules have 3D coordinates, proteins have orientations, and physical forces transform predictably under rotation. Geometric deep learning addresses this by building symmetries directly into model architectures. Notes in this section cover equivariant networks, focusing on SO(3) and SE(3) equivariance achieved through group representation theory and spherical harmonics. The goal is to understand how these models work and why the symmetry constraints matter for data efficiency and physical correctness.

Year	Paper	Key Idea
2018	3D Steerable CNNs: Rotationally Equivariant Features	SE(3)-equivariant features via Wigner D-matrices and spherical harmonics
2018	Defining Disentangled Representations via Group Theory	First formal definition of disentanglement using symmetry group decomposition
2018	Spherical CNNs: Rotation-Equivariant Networks on the Sphere	SO(3)-equivariance via generalized Fourier transform on the sphere
2019	DGCNN: Dynamic Graph CNN for Point Clouds	EdgeConv on dynamically recomputed k-NN graphs in feature space
2020	SE(3)-Transformers: Equivariant Attention for 3D Data	Self-attention with SE(3)-equivariant type-0/type-1 features

Machine Learning

Three-panel diagram showing symmetry group decomposition, equivariant mapping from world states to representations, and block-diagonal disentangled decomposition

Defining Disentangled Representations via Group Theory

Proposes the first principled mathematical definition of disentangled representations by connecting symmetry group decompositions to independent subspaces in a representation’s vector space.

Machine Learning

Three-panel diagram showing DGCNN point cloud processing: input space k-NN graph, EdgeConv operation, and semantic feature space clustering

DGCNN: Dynamic Graph CNN for Point Cloud Learning

DGCNN introduces the EdgeConv operator, which constructs k-nearest neighbor graphs dynamically in feature space at each network layer. This enables the model to capture both local geometry and long-range semantic relationships for point cloud classification and segmentation.

Machine Learning

SE(3)-Transformer architecture showing invariant attention weights modulating equivariant value messages on a 3D point cloud

SE(3)-Transformers: Equivariant Attention for 3D Data

Fuchs et al. introduce the SE(3)-Transformer, which combines self-attention with SE(3)-equivariance for 3D point clouds and graphs. Invariant attention weights modulate equivariant value messages from tensor field networks, resolving angular filter constraints while enabling data-adaptive, anisotropic processing.

Machine Learning

Comparison of planar CNN (translation only) versus spherical CNN (SO(3)-equivariant) showing how filters rotate on the sphere

Spherical CNNs: Rotation-Equivariant Networks on the Sphere

Cohen et al. introduce Spherical CNNs that achieve SO(3)-equivariance by defining cross-correlation on the sphere and rotation group, computed efficiently via generalized FFT algorithms from non-commutative harmonic analysis.

Machine Learning

Comparison of standard 3D CNN versus 3D Steerable CNN for handling rotational symmetry

3D Steerable CNNs: Rotationally Equivariant Features

Weiler et al.’s NeurIPS 2018 paper introducing 3D Steerable CNNs that achieve SE(3) equivariance through group representation theory and spherical harmonic convolution kernels, eliminating the need for rotational data augmentation and improving data efficiency for scientific applications with rotational symmetry like molecular and protein structures.