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

2024-01-25

arXiv.org

AI Summary

Modeling graph-structured data residing on Riemannian manifolds poses challenges in preserving intrinsic geometric structure while ensuring equivariance under both node permutations and manifold isometries. Method: We propose the first equivariant graph neural network layer that jointly incorporates manifold diffusion modeling and nonlinear equivariant mapping in tangent spaces. The layer defines graph convolution via the manifold diffusion equation and constructs an equivariant multilayer perceptron in the tangent space at each node, enabling native support for arbitrary graph topologies and sizes. Contribution/Results: Our layer rigorously satisfies equivariance under node permutations and Riemannian isometries, and uniformly accommodates diverse Riemannian manifolds—including spheres, hyperbolic spaces, and triangulated surfaces—by embedding strong geometric inductive biases. Experiments on synthetic manifold graph datasets and a real-world Alzheimer’s disease classification task using right hippocampal triangular meshes demonstrate performance competitive with or superior to state-of-the-art specialized methods, alongside significantly improved generalization.

Technology Category

• Machine Learning: Learning with Manifolds
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Computer Vision: Diffusion Models for Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

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.

Problem

• Develops GCN for Riemannian manifold graphs
• Creates diffusion and tangent MLP layers
• Enhances Alzheimer's classification on hippocampus meshes

Innovation

• Manifold-valued graph diffusion
• Tangent multilayer perceptron
• Equivariant under node permutations