Learning a Gaussian Process Model on the Riemannian Manifold of Non-decreasing Distribution Functions

Chafik Samir (1) , Jean-Michel Loubes (2) , Anne-Françoise Yao (1) , François Bachoc (2)

1 UCA [2017-2020] - Université Clermont Auvergne [2017-2020]
2 IMT - Institut de Mathématiques de Toulouse UMR5219

In this work, we consider the problem of learning regression models from a finite set of functional objects. In particular, we introduce a novel framework to learn a Gaussian process model on the space of Strictly Non-decreasing Distribution Functions (SNDF). Gaussian processes (GPs) are commonly known to provide powerful tools for non-parametric regression and uncertainty estimation on vector spaces. On top of that, we define a Riemannian structure of the SNDF space and we learn a GP model indexed by SNDF. Such formulation enables to define an appropriate covariance function, extending the Matérn family of covariance functions. We also show how the full Gaussian process methodology, namely covariance parameter estimation and prediction, can be put into action on the SNDF space. The proposed method is tested using multiple simulations and validated on real-world data.

Mots clés: Gaussian process; Riemannian manifold; Functional data

Domaines: Statistiques [math.ST]

Chafik Samir, Jean-Michel Loubes, Anne-Françoise Yao, François Bachoc. Learning a Gaussian Process Model on the Riemannian Manifold of Non-decreasing Distribution Functions. 16th Pacific Rim International Conference on Artificial Intelligence, Aug 2019, Cuvu, Fiji. ⟨10.1007/978-3-030-29911-8_9⟩. ⟨hal-02276949⟩