Wasserstein-type Gaussian Process Regressions for Input Measurement Uncertainty

TL;DR

This paper introduces a Wasserstein-based Gaussian process regression method that effectively handles input measurement errors, providing more accurate uncertainty quantification without complex latent variable models.

Contribution

It proposes a novel PWA kernel for GPs that directly models input uncertainty as probability measures, improving robustness and transparency over existing methods.

Findings

PWA kernels admit closed-form expressions for 1D components.

PWAGPs handle input noise without unobserved covariates.

The method improves uncertainty quantification in noisy input scenarios.

Abstract

Gaussian process (GP) regression is widely used for uncertainty quantification, yet the standard formulation assumes noise-free covariates. When inputs are measured with error, this errors-in-variables (EIV) setting can lead to optimistically narrow posterior intervals and biased decisions. We study GP regression under input measurement uncertainty by representing each noisy input as a probability measure and defining covariance through Wasserstein distances between these measures. Building on this perspective, we instantiate a deterministic projected Wasserstein ARD (PWA) kernel whose one-dimensional components admit closed-form expressions and whose product structure yields a scalable, positive-definite kernel on distributions. Unlike latent-input GP models, PWA-based GPs (\PWAGPs) handle input noise without introducing unobserved covariates or Monte Carlo projections, making uncertainty quantification more transparent and robust.