Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Martin Hanik, Gabriele Steidl, Christoph von Tycowicz

TL;DR

This work introduces two graph neural network layers tailored for manifold-valued node features: a diffusion layer grounded in the manifold diffusion equation and a tangent multilayer perceptron (tMLP). Both layers are equivariant to node permutations and to the isometries of the feature manifold, imparting a useful inductive bias for learning on non-Euclidean domains. The authors show that a generic Manifold GCN block integrating these layers performs competitively with task-specific architectures on synthetic graph classification and Alzheimer's disease classification from hippocampal meshes, often with fewer parameters and better data efficiency. The approach enables processing of diverse manifolds (e.g., hyperbolic, SPD, spheres) and is released in the Morphomatics library to facilitate broad adoption in geometric deep learning and medical shape analysis.

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.

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

TL;DR

This work introduces two graph neural network layers tailored for manifold-valued node features: a diffusion layer grounded in the manifold diffusion equation and a tangent multilayer perceptron (tMLP). Both layers are equivariant to node permutations and to the isometries of the feature manifold, imparting a useful inductive bias for learning on non-Euclidean domains. The authors show that a generic Manifold GCN block integrating these layers performs competitively with task-specific architectures on synthetic graph classification and Alzheimer's disease classification from hippocampal meshes, often with fewer parameters and better data efficiency. The approach enables processing of diverse manifolds (e.g., hyperbolic, SPD, spheres) and is released in the Morphomatics library to facilitate broad adoption in geometric deep learning and medical shape analysis.

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.
Paper Structure (36 sections, 8 theorems, 43 equations, 4 figures, 4 tables)

This paper contains 36 sections, 8 theorems, 43 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Flow-based GNNs in the Euclidean Space
  4. Diffusion networks
  5. Neural ordinary differential equations
  6. Deep Neural Networks for Manifold-valued Signals
  7. Networks for manifold-valued grids
  8. GNNs for special manifolds
  9. Flow-based networks on manifolds
  10. Background
  11. Riemannian Geometry
  12. Graph Laplacian in Manifolds
  13. Equivariance and Invariance
  14. Diffusion Layer
  15. Tangent Multilayer Perceptron
  16. ...and 21 more sections

Key Result

Theorem 1

Let $G = (V, E, w, f)$ be a graph with positive weights and features $f \in \mathcal{H}(V, M)$ such that the smallest closed geodesic ball that contains the graph's features is convex. Assume $\sum_{u \sim v} w(v, u) \le 1$ for all $v \in V$. Then, Equation (eq:diffusion) has a solution $\widetilde{

Figures (4)

  • Figure 1: Diffusion (indicated by color) of the vertices of a rectangle graph on the 2-sphere
  • Figure 2: Graphs created with the Erdős-Renyi (left), Barabasi-Albert (middle), and Watts-Strogatz (right) algorithms
  • Figure 3: Hippocampus mesh with vertex normals (left) and the same normals, colored like their footpoints, as points on the 2-sphere (right)
  • Figure 4: Stable tetrahedron-graph on the 2-sphere

Theorems & Definitions (20)

  • Theorem 1
  • Remark 1
  • Definition 1: diffusion layer
  • Theorem 2
  • proof
  • Definition 2
  • Remark 2
  • Theorem 3
  • proof
  • Lemma 4
  • ...and 10 more