Manifold GCN: Diffusion-based Convolutional Neural Network for Manifold-valued Graphs

TL;DR

This paper introduces two novel graph neural network layers designed for graphs with features on Riemannian manifolds, leveraging diffusion processes and tangent space modeling to improve performance on manifold-valued data.

Contribution

It presents a diffusion-based layer and a tangent multilayer perceptron for manifold-valued graphs, with permutation and isometry equivariance, broadening GNN applicability.

Findings

Effective on synthetic and real data, including Alzheimer's classification.

Perform as well as or better than state-of-the-art task-specific networks.

Applicable to a wide range of manifold-valued graph problems.

Abstract

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.