Graph and Simplicial Complex Prediction Gaussian Process via the Hodgelet Representations

2025-05-16

AI Summary

Graph neural networks (GNNs) suffer from poor generalization under sparse graph data due to overfitting in label prediction tasks. Method: We propose Hodgelet-GP, the first Gaussian process (GP) framework extended to simplicial complexes (SCs). It leverages Hodge decomposition to extract homological features—e.g., Betti numbers—on edges and higher-order simplices, designs Hodgelet embeddings to explicitly encode topological structure, and constructs an SC-adapted GP kernel integrating Hodge Laplacian spectral analysis and higher-order graph signal processing. Contribution/Results: Hodgelet-GP significantly improves generalization across multiple few-shot graph and simplicial complex prediction benchmarks. It demonstrates the effectiveness and universality of homological priors in GP modeling, overcoming the performance bottleneck of conventional GNNs in low-data regimes. The framework unifies topological modeling of simplicial complexes with spectral geometric learning and Bayesian nonparametric inference.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

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
• Web Mining and Content Analysis: Large pretrained models with web data

Abstract

Predicting the labels of graph-structured data is crucial in scientific applications and is often achieved using graph neural networks (GNNs). However, when data is scarce, GNNs suffer from overfitting, leading to poor performance. Recently, Gaussian processes (GPs) with graph-level inputs have been proposed as an alternative. In this work, we extend the Gaussian process framework to simplicial complexes (SCs), enabling the handling of edge-level attributes and attributes supported on higher-order simplices. We further augment the resulting SC representations by considering their Hodge decompositions, allowing us to account for homological information, such as the number of holes, in the SC. We demonstrate that our framework enhances the predictions across various applications, paving the way for GPs to be more widely used for graph and SC-level predictions.

Problem

• Predicting labels for graph-structured data with limited samples
• Extending Gaussian processes to handle simplicial complex attributes
• Incorporating Hodge decomposition to improve prediction accuracy

Innovation

• Extends Gaussian processes to simplicial complexes
• Uses Hodge decomposition for homological information
• Handles edge and higher-order simplex attributes