A Gaussian Process Regression Model for Distribution Inputs

François Bachoc (1, 2) , Fabrice Gamboa (2, 1) , Jean-Michel Loubes (2) , Nil Venet (3, 2)

1 GdR MASCOT-NUM - Méthodes d'Analyse Stochastique des Codes et Traitements Numériques
2 IMT - Institut de Mathématiques de Toulouse UMR5219
3 CEA - Commissariat à l'énergie atomique et aux énergies alternatives

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.

Mots clés: Kriging; Positive definite kernel; Gaussian process; Index Terms—Gaussian process; Fractional Brownian motion; Krig- ing; Wasserstein distance; Monge-Kantorovich distance

Domaines: Machine Learning [stat.ML]; Théorie [stat.TH]; Probabilités [math.PR]

François Bachoc, Fabrice Gamboa, Jean-Michel Loubes, Nil Venet. A Gaussian Process Regression Model for Distribution Inputs. IEEE Transactions on Information Theory, 2018, 64 (10), pp.6620 - 6637. ⟨hal-01450002v2⟩