Isotropic Gaussian Processes on Finite Spaces of Graphs

Viacheslav Borovitskiy, Mohammad Reza Karimi, Vignesh Ram Somnath, Andreas Krause
Proceedings of The 26th International Conference on Artificial Intelligence and Statistics, PMLR 206:4556-4574, 2023.

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 Matérn. 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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v206-borovitskiy23a, title = {Isotropic Gaussian Processes on Finite Spaces of Graphs}, author = {Borovitskiy, Viacheslav and Karimi, Mohammad Reza and Somnath, Vignesh Ram and Krause, Andreas}, booktitle = {Proceedings of The 26th International Conference on Artificial Intelligence and Statistics}, pages = {4556--4574}, year = {2023}, editor = {Ruiz, Francisco and Dy, Jennifer and van de Meent, Jan-Willem}, volume = {206}, series = {Proceedings of Machine Learning Research}, month = {25--27 Apr}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v206/borovitskiy23a/borovitskiy23a.pdf}, url = {https://proceedings.mlr.press/v206/borovitskiy23a.html}, 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 Matérn. 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.} }
Endnote
%0 Conference Paper %T Isotropic Gaussian Processes on Finite Spaces of Graphs %A Viacheslav Borovitskiy %A Mohammad Reza Karimi %A Vignesh Ram Somnath %A Andreas Krause %B Proceedings of The 26th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2023 %E Francisco Ruiz %E Jennifer Dy %E Jan-Willem van de Meent %F pmlr-v206-borovitskiy23a %I PMLR %P 4556--4574 %U https://proceedings.mlr.press/v206/borovitskiy23a.html %V 206 %X 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 Matérn. 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.
APA
Borovitskiy, V., Karimi, M.R., Somnath, V.R. & Krause, A.. (2023). Isotropic Gaussian Processes on Finite Spaces of Graphs. Proceedings of The 26th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 206:4556-4574 Available from https://proceedings.mlr.press/v206/borovitskiy23a.html.

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