Gaussian Process regression over discrete probability measures: on the non-stationarity relation between Euclidean and Wasserstein Squared Exponential Kernels

Gaussian Process regression is a kernel method successfully adopted in many real-life applications. Recently, there is a growing interest on extending this method to non-Euclidean input spaces, like the one considered in this paper, consisting of probability measures. Although a Positive Definite ke...

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Bibliographic Details
Published in:Journal of global optimization Vol. 92; no. 2; pp. 253 - 278
Main Authors: Candelieri, Antonio, Ponti, Andrea, Archetti, Francesco
Format: Journal Article
Language:English
Published: New York Springer US 01.06.2025
Springer Nature B.V
Subjects:
Algebra
Computer Science
Euclidean geometry
Gaussian process
Kernel functions
Mathematics
Mathematics and Statistics
Neural networks
Operations Research/Decision Theory
Optimization
Real Functions
Riemann manifold
Statistical analysis
Gaussian Process
Wasserstein
Optimal transport
Kernel
Non-stationarity
ISSN:0925-5001, 1573-2916
Online Access:Get full text
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Summary:Gaussian Process regression is a kernel method successfully adopted in many real-life applications. Recently, there is a growing interest on extending this method to non-Euclidean input spaces, like the one considered in this paper, consisting of probability measures. Although a Positive Definite kernel can be defined by using a suitable distance—the Wasserstein distance— the common procedure for learning the Gaussian Process model can fail due to numerical issues, arising earlier and more frequently than in the case of an Euclidean input space and, as demonstrated, impossible to avoid by adding artificial noise ( nugget effect ) as usually done. This paper uncovers the main reason of these issues, that is a non-stationarity relation between the Wasserstein-based squared exponential kernel and its Euclidean counterpart. As a relevant result, we learn a Gaussian Process model by assuming the input space as Euclidean and then use an algebraic transformation, based on the uncovered relation, to transform it into a non-stationary and Wasserstein-based Gaussian Process model over probability measures. This algebraic transformation is simpler than log-exp maps used on data belonging to Riemannian manifolds and recently extended to consider the pseudo-Riemannian structure of an input space equipped with the Wasserstein distance.
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ISSN:0925-5001
1573-2916
DOI:10.1007/s10898-025-01463-y