Gaussian Processes on Distributions based on Regularized Optimal Transport

François Bachoc, Louis Béthune, Alberto Gonzalez-Sanz, Jean-Michel Loubes
Proceedings of The 26th International Conference on Artificial Intelligence and Statistics, PMLR 206:4986-5010, 2023.

Abstract

We present a novel kernel over the space of probability measures based on the dual formulation of optimal regularized transport. We propose an Hilbertian embedding of the space of probabilities using their Sinkhorn potentials, which are solutions of the dual entropic relaxed optimal transport between the probabilities and a reference measure $\mathcal{U}$. We prove that this construction enables to obtain a valid kernel, by using the Hilbert norms. We prove that the kernel enjoys theoretical properties such as universality and some invariances, while still being computationally feasible. Moreover we provide theoretical guarantees on the behaviour of a Gaussian process based on this kernel. The empirical performances are compared with other traditional choices of kernels for processes indexed on distributions.

Cite this Paper

BibTeX
@InProceedings{pmlr-v206-bachoc23a, title = {Gaussian Processes on Distributions based on Regularized Optimal Transport}, author = {Bachoc, Fran\c{c}ois and B\'ethune, Louis and Gonzalez-Sanz, Alberto and Loubes, Jean-Michel}, booktitle = {Proceedings of The 26th International Conference on Artificial Intelligence and Statistics}, pages = {4986--5010}, 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/bachoc23a/bachoc23a.pdf}, url = {https://proceedings.mlr.press/v206/bachoc23a.html}, abstract = {We present a novel kernel over the space of probability measures based on the dual formulation of optimal regularized transport. We propose an Hilbertian embedding of the space of probabilities using their Sinkhorn potentials, which are solutions of the dual entropic relaxed optimal transport between the probabilities and a reference measure $\mathcal{U}$. We prove that this construction enables to obtain a valid kernel, by using the Hilbert norms. We prove that the kernel enjoys theoretical properties such as universality and some invariances, while still being computationally feasible. Moreover we provide theoretical guarantees on the behaviour of a Gaussian process based on this kernel. The empirical performances are compared with other traditional choices of kernels for processes indexed on distributions.} }
Endnote
%0 Conference Paper %T Gaussian Processes on Distributions based on Regularized Optimal Transport %A François Bachoc %A Louis Béthune %A Alberto Gonzalez-Sanz %A Jean-Michel Loubes %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-bachoc23a %I PMLR %P 4986--5010 %U https://proceedings.mlr.press/v206/bachoc23a.html %V 206 %X We present a novel kernel over the space of probability measures based on the dual formulation of optimal regularized transport. We propose an Hilbertian embedding of the space of probabilities using their Sinkhorn potentials, which are solutions of the dual entropic relaxed optimal transport between the probabilities and a reference measure $\mathcal{U}$. We prove that this construction enables to obtain a valid kernel, by using the Hilbert norms. We prove that the kernel enjoys theoretical properties such as universality and some invariances, while still being computationally feasible. Moreover we provide theoretical guarantees on the behaviour of a Gaussian process based on this kernel. The empirical performances are compared with other traditional choices of kernels for processes indexed on distributions.
APA
Bachoc, F., Béthune, L., Gonzalez-Sanz, A. & Loubes, J.. (2023). Gaussian Processes on Distributions based on Regularized Optimal Transport. Proceedings of The 26th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 206:4986-5010 Available from https://proceedings.mlr.press/v206/bachoc23a.html.

Related Material