# Wasserstein-type Gaussian Process Regressions for Input Measurement Uncertainty

arXiv: https://arxiv.org/abs/2603.17271 · HTML: https://mlsanity.com/paper/2603.17271/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 45 references that are in the map.
- First posted on arXiv 18 Mar 2026.

## Builds on (most cited first, 15 of 15)

- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Statistical Aspects of Wasserstein Distances](https://mlsanity.com/paper/1806.05500/) (2018)
- [Kernel Interpolation for Scalable Structured Gaussian Processes (KISS-GP)](https://mlsanity.com/paper/1503.01057/) (2015)
- [Uniform Error Bounds for Gaussian Process Regression with Application to Safe Control](https://mlsanity.com/paper/1906.01376/) (2019)
- [Learning Theory for Distribution Regression](https://mlsanity.com/paper/1411.2066/) (2014)
- [Heterogeneous Multi-output Gaussian Process Prediction](https://mlsanity.com/paper/1805.07633/) (2018)
- [Sliced-Wasserstein on Symmetric Positive Definite Matrices for M/EEG Signals](https://mlsanity.com/paper/2303.05798/) (2023)
- [Distribution Regression with Sliced Wasserstein Kernels](https://mlsanity.com/paper/2202.03926/) (2022)
- [A Gaussian Process Regression Model for Distribution Inputs](https://mlsanity.com/paper/1701.09055/) (2017)
- [Decompositions of the mean continuous ranked probability score](https://mlsanity.com/paper/2311.14122/) (2023)
- [Uncertainty Quantification for Bayesian Optimization](https://mlsanity.com/paper/2002.01569/) (2020)
- [Errors-in-variables Fr'echet Regression with Low-rank Covariate Approximation](https://mlsanity.com/paper/2305.09282/) (2023)
- [Gaussian Process regression over discrete probability measures: on the non-stationarity relation between Euclidean and Wasserstein Squared Exponential Kernels](https://mlsanity.com/paper/2212.01310/) (2022)
- [Wavelet Tree Ensembles for Triangulable Manifolds](https://mlsanity.com/paper/2601.20254/) (2026)
- [GGMPs: Generalized Gaussian Mixture Processes](https://mlsanity.com/paper/2603.10442/) (2026)

## Abstract (arXiv)

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.