[Paper Review] Graph Convolutional Gaussian Processes

Ian D. Walker, Ben Glocker|arXiv (Cornell University)|May 14, 2019

Gaussian Processes and Bayesian Inference9 citations

TL;DR

This paper introduces Graph Convolutional Gaussian Processes (GCGP), a Bayesian nonparametric method that extends convolutional inductive biases to non-Euclidean graphs using Gaussian process priors. By modeling local patch responses via GPs and aggregating them across the graph, GCGP achieves high expressivity with reduced dimensionality, outperforming deep learning baselines on small datasets in image and mesh classification tasks while providing uncertainty estimates.

ABSTRACT

We propose a novel Bayesian nonparametric method to learn translation-invariant relationships on non-Euclidean domains. The resulting graph convolutional Gaussian processes can be applied to problems in machine learning for which the input observations are functions with domains on general graphs. The structure of these models allows for high dimensional inputs while retaining expressibility, as is the case with convolutional neural networks. We present applications of graph convolutional Gaussian processes to images and triangular meshes, demonstrating their versatility and effectiveness, comparing favorably to existing methods, despite being relatively simple models.

Motivation & Objective

• To develop a Bayesian, nonparametric method that learns translation-invariant relationships on general graphs, extending convolutional inductive biases beyond Euclidean domains.
• To address the curse of dimensionality in standard Gaussian processes by focusing on local graph neighborhoods through a patch-based decomposition.
• To enable uncertainty-aware learning on high-dimensional inputs defined on non-Euclidean graphs such as images, triangular meshes, and social networks.
• To provide a principled, scalable alternative to deep graph neural networks that performs well even with limited training data.
• To lay the foundation for hierarchical and flexible GP models on graphs with variable structure and vertex counts.

Proposed method

• The method decomposes the input signal into local graph neighborhoods (patches) and models each patch’s response using a shared Gaussian process prior.
• The overall function is defined as the sum of patch responses, inducing a GP prior on the full input via additive GP structure.
• A stationary kernel (e.g., RBF) is used for the patch response function $ g $, ensuring translation invariance across the graph.
• The model is trained using variational inference, with a structured approximation to the GP posterior to enable scalable inference.
• The approach is applied to both Euclidean grids (images) and non-Euclidean graphs (triangular meshes, superpixel graphs), with consistent performance across domains.
• The method assumes fixed vertex count per graph, enabling consistent input dimensionality while allowing variable edge structures.

Experimental results

Research questions

• RQ1 Can convolutional inductive biases be effectively extended from Euclidean domains (e.g., images) to general non-Euclidean graphs using Gaussian processes?
• RQ2 Can a GP-based model maintain expressivity while reducing dimensionality through local patch aggregation on graphs?
• RQ3 Does the combination of GP uncertainty and convolutional invariance improve generalization on small datasets compared to deep learning baselines?
• RQ4 How does the model perform on non-Euclidean domains such as 3D meshes and superpixel graphs, where standard CNNs are inapplicable?
• RQ5 Can the model be extended to handle graphs with variable vertex counts, and what are the limitations of the current fixed-vertex framework?

Key findings

• GCGP achieves competitive classification accuracy on MNIST superpixel and 3D mesh datasets, outperforming deep learning models like MoNet that overfit due to limited training data.
• On the 2500-vertex mesh classification task, the GCGP model achieved significantly better performance than a one-layer MoNet, which reached only 33.33% accuracy.
• The model consistently confuses visually similar individuals (e.g., subjects 4 and 8 from MPI Faust) at low resolution, indicating that performance is limited by local feature similarity.
• Despite being a shallow, wide model, GCGP demonstrates strong generalization on small datasets, suggesting effective inductive bias and data efficiency.
• The Bayesian nature of GCGP provides meaningful uncertainty estimates, a key advantage over standard deep learning models.
• The method shows promise as a foundation for future hierarchical models, especially when combined with deep GP frameworks.