A Gaussian Process Regression Model for Distribution Inputs

Abstract--Monge-Kantorovich distances, otherwise known as Wasserstein distances, have received a growing attention in statistics and machine learning as a powerful discrepancy measure for probability distributions. In this paper, we focus on forecasting a Gaussian process indexed by probability distributions. For this, we provide a family of positive definite kernels built using transportation based distances. We provide a probabilistic understanding of these kernels and characterize the corresponding stochastic processes. We prove that the Gaussian processes indexed by distributions corresponding to these kernels can be efficiently forecast, opening new perspectives in Gaussian process modeling. RIGINALLY used in spatial statistics (see for instance [2] and references therein), Kriging has become very popular in many fields such as machine learning or computer experiment, as described in [3]. It consists in predicting the value of a function at some point by a linear combination of observed values at different points. The unknown function is modeled as the realization of a random process, usually Gaussian, and the Kriging forecast can be seen as the posterior mean, leading to the optimal linear unbiased predictor of the random process. Gaussian process models rely on the definition of a covariance function that characterizes the correlations between values of the process at different observation points. As the notion of similarity between data points is crucial, i.e. close location inputs are likely to have similar target values, covariance functions are the key ingredient in using Gaussian processes, since they define nearness or similarity.