Isotropic Gaussian Processes on Finite Spaces of Graphs

TL;DR

This paper introduces a method to define and compute Gaussian process priors on various graph sets, respecting their geometric structure, and demonstrates their application in molecular property prediction.

Contribution

It presents a novel framework for Gaussian processes on graph spaces, including efficient kernel computation and handling of graph equivalence classes.

Findings

Efficient kernel evaluation for graph Gaussian processes.

Application to molecular property prediction with small data.

Approximation methods for intractable kernel computations.

Abstract

We propose a principled way to define Gaussian process priors on various sets of unweighted graphs: directed or undirected, with or without loops. We endow each of these sets with a geometric structure, inducing the notions of closeness and symmetries, by turning them into a vertex set of an appropriate metagraph. Building on this, we describe the class of priors that respect this structure and are analogous to the Euclidean isotropic processes, like squared exponential or Mat\'ern. We propose an efficient computational technique for the ostensibly intractable problem of evaluating these priors' kernels, making such Gaussian processes usable within the usual toolboxes and downstream applications. We go further to consider sets of equivalence classes of unweighted graphs and define the appropriate versions of priors thereon. We prove a hardness result, showing that in this case, exact kernel computation cannot be performed efficiently. However, we propose a simple Monte Carlo approximation for handling moderately sized cases. Inspired by applications in chemistry, we illustrate the proposed techniques on a real molecular property prediction task in the small data regime.