Graph and Simplicial Complex Prediction Gaussian Process via the Hodgelet Representations

TL;DR

This paper extends Gaussian processes to simplicial complexes, incorporating Hodge decompositions to improve graph and higher-order structure predictions, especially in data-scarce scenarios.

Contribution

The work introduces a novel GP framework for simplicial complexes with Hodge-based augmentation, capturing homological features for enhanced prediction accuracy.

Findings

Improved prediction accuracy on graph and simplicial complex data

Effective incorporation of homological information like holes

Enhanced performance in low-data regimes

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.