Manifold GCN: Diffusion-based Convolutional Neural Network for Manifold-valued Graphs
This work addresses a domain-specific problem for researchers and practitioners in fields like medical imaging and geometry processing, offering a more general approach for manifold-valued graphs.
The authors tackled the problem of graph neural networks for data on Riemannian manifolds by proposing two equivariant layers based on manifold-valued graph diffusion and tangent multilayer perceptrons, achieving performance comparable to or better than task-specific state-of-the-art networks in synthetic data and Alzheimer's classification on hippocampus meshes.
We propose two graph neural network layers for graphs with features in a Riemannian manifold. First, based on a manifold-valued graph diffusion equation, we construct a diffusion layer that can be applied to an arbitrary number of nodes and graph connectivity patterns. Second, we model a tangent multilayer perceptron by transferring ideas from the vector neuron framework to our general setting. Both layers are equivariant under node permutations and the feature manifold's isometries. These properties have led to a beneficial inductive bias in many deep-learning tasks. Numerical examples on synthetic data and an Alzheimer's classification application on triangle meshes of the right hippocampus demonstrate the usefulness of our new layers: While they apply to a much broader class of problems, they perform as well as or better than task-specific state-of-the-art networks.