A Gaussian Process Regression Model for Distribution Inputs
This provides a novel method for Gaussian process modeling with distribution inputs, addressing a specific bottleneck in handling probability distributions as inputs.
The paper tackles the problem of forecasting Gaussian processes indexed by probability distributions by developing positive definite kernels based on Wasserstein distances, proving these kernels enable efficient forecasting of the corresponding Gaussian processes.
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.