Gaussian Processes on Cellular Complexes

TL;DR

This paper introduces Gaussian processes on cellular complexes, extending graph-based models to capture higher-order interactions among vertices, edges, and cells, with new kernels that generalize existing graph kernels.

Contribution

It proposes Gaussian processes on cellular complexes and derives two novel kernels, advancing the modeling of complex topological interactions in machine learning.

Findings

Derived a kernel generalizing the graph Matérn kernel

Developed a kernel that mixes information across cell types

Enhanced modeling of higher-order relations in data

Abstract

In recent years, there has been considerable interest in developing machine learning models on graphs to account for topological inductive biases. In particular, recent attention has been given to Gaussian processes on such structures since they can additionally account for uncertainty. However, graphs are limited to modelling relations between two vertices. In this paper, we go beyond this dyadic setting and consider polyadic relations that include interactions between vertices, edges and one of their generalisations, known as cells. Specifically, we propose Gaussian processes on cellular complexes, a generalisation of graphs that captures interactions between these higher-order cells. One of our key contributions is the derivation of two novel kernels, one that generalises the graph Mat\'ern kernel and one that additionally mixes information of different cell types.