Wasserstein-type Gaussian Process Regressions for Input Measurement Uncertainty

Hengrui Luo; Xiaoye S. Li; Yang Liu; Marcus Noack; Ji Qiang; Mark D. Risser

stat.MEMar 18, 2026

Wasserstein-type Gaussian Process Regressions for Input Measurement Uncertainty

Authors: Hengrui Luo, Xiaoye S. Li, Yang Liu, Marcus Noack, Ji Qiang, Mark D. Risser

Organizations: Department of Statistics Rice University, Houston, TX 77005 · Lawrence Berkeley National Laboratory, Berkeley, CA 94709

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.

Figures & tables

Method (paper name)	Kernel input object	Dist.	Latent $X$	Det.	EIV mechanism	Citations
Regular GP (RBF / Matérn / Exp)	Point estimate (empirical mean $m_{i}$ )	N	N	Y	Ignores input uncertainty (variance / shape); standard baseline in Sec. 3 and Table 1.	Williams and Rasmussen, (2006)
Aggregated Regular GPs	Noisy samples $\{U_{ij}\}$ as point inputs (fit per- $j$ GP; average)	N	N	Y	Ensemble over replicate-specific GPs; can amplify variance in sparse regions (as observed).	This paper.
IV-function / cov-only encoding	Gaussian summary $(\mu_{x},\Sigma_{x})$ but only $\Sigma_{x}$ used in kernel	Y*	N	Y	Efficient for locally Gaussian perturbations; cannot represent skew / multimodality / complex support.	Moreno-Muñoz et al., (2018)
WGP (Eq. (5), WGP-RBF)	Measure $\mu$	Y	N	Y	OT-based kernel on distributions; PD only in special cases (e.g., Gaussian $p{=}2$ , or 1D).	Candelieri et al., (2022) ; Peyré et al., (2019)
sliced WGP (Eq. (6), SWGP)	Measure $\mu$ ; average $W_{p}$ over random 1D projections $u$	Y	N	MC	PD preserved via expectation over projections; stochastic projections add Monte Carlo variability and cost.	Meunier et al., (2022) ; Bonet et al., (2023)
PWA-GP (Eq. (7))	Coordinate-wise 1D marginals $\{\mu_{i}\}_{i=1}^{d}$	Y	N	Y	Product of 1D Wasserstein kernels with dimension-specific hyperparameters (ARD-like); PD via product closure.	This paper.

Table 1: Compact comparison of EIV / uncertain-input methods mentioned in the manuscript (including experimental baselines and Wasserstein variants) plus an added distribution-input competitor family (KME/MMD). Dist. indicates whether the model explicitly consumes a distribution/set input (Y* = Gaussian-summary distribution but only covariance used). Det.: deterministic vs Monte Carlo (MC) due to random projections. OT = optimal transport; PD = positive definite; KME = kernel mean embedding; MMD = maximum mean discrepancy; PCA = principal component analysis.

Panel of Figure 1 — Figure 1: Illustration of error-in-variable regression problem. In both panels, the true function is $y=f(X)=\frac{\sin(10\pi\cdot X)}{2X}+(X-1)^{4}$ , but in each case the “true” input locations $X$ are contaminated with measurement errors (with standard deviation of 0.01 and 0.05 on the left and right, respectively). Using a GP to accurately infer the true function (the blue line) must account for the fact that the input locations $U$ are uncertain.

Scenario	Metric	Reg	Agg	WGP	SWGP	PWA	PCPWA	UI	KME	MMD
2D-mean	RMSE	8.500e-03	1.061e-01	5.280e-02	1.015e+07	4.620e-02	1.615e-01	4.000e-04	2.100e-03	2.600e-03
	Cov.	4.250e-01	1.000e-01	1.000e+00	0.000e+00	6.250e-01	9.750e-01	1.000e+00	1.000e+00	9.750e-01
	CRPS	5.100e-03	5.940e-02	4.540e-02	7.752e+06	3.970e-02	1.108e-01	2.000e-03	8.900e-03	1.010e-02
1D-EIV	RMSE	3.090e-01	5.712e-01	3.014e-01	2.995e-01	3.014e-01	3.014e-01	3.462e-01	4.179e-01	3.029e-01
	Cov.	6.500e-01	5.000e-02	9.670e-01	9.670e-01	9.670e-01	9.670e-01	5.670e-01	5.330e-01	6.670e-01
	CRPS	1.848e-01	4.791e-01	1.716e-01	1.722e-01	1.716e-01	1.716e-01	2.179e-01	2.844e-01	1.775e-01

Table 2: Simulated distributional regression experiments. For each scenario we report test RMSE, empirical coverage of nominal

90\%

predictive intervals (Cov.), and CRPS. “Reg” = Regular GP on empirical means, “Agg” = Aggregated GP, “UI” = uncertain-input GP. Extremely large SWGP values indicate numerical divergence. Values of Cov. above

0.9

are shown in bold .

	5% error			0.5% error
Method	RMSE	Cov.	CRPS	RMSE	Cov.	CRPS
RBF GP	0.2186	0.910	0.0883	0.2177	0.920	0.0806
Matérn $3/2$ GP	0.2351	0.930	0.1010	0.2378	0.930	0.0997
Matérn $5/2$ GP	0.2228	0.940	0.0912	0.2242	0.940	0.0892
Exp. GP	0.2432	0.980	0.1380	0.2847	0.980	0.1392
PWA-GP	0.2798	0.980	0.1690	0.3373	0.980	0.1681

Table 3: Accelerator dataset: test RMSE, empirical coverage of nominal

90\%

predictive intervals (Cov.), and CRPS for two relative error levels. Euclidean GPs use empirical means as inputs; distributional GPs act on full input distributions.

Panel of Figure 2 — Figure 2: RMSE (left) and CRPS (right) for GP-RBF vs. WGP -RBF across training/testing years 1987-2022. GP-RBF degrades when trained on recent and tested on older years, indicating poor generalization under temporal shifts. WGP -RBF remains robust, consistently achieving lower errors and better probabilistic calibration.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Method	Effective # inputs	Kernel assembly cost (big-O)
Regular GP on means ( Williams and Rasmussen,, 2006 )	$N$	$\mathcal{O}(N^{2}d)$
Aggregated GP on samples ( Williams and Rasmussen,, 2006 )	$NM$	$\mathcal{O}(N^{2}M^{2}d)$ (matrix) $+$ $\mathcal{O}(N^{3}M^{3})$ (training)
Uncertain-input GP ( Girard et al.,, 2002 ; McHutchon and Rasmussen,, 2011 )	$N$	$\mathcal{O}(N^{2}d^{3})$ (full cov.) or $\mathcal{O}(N^{2}d)$ (diag)
Full WGP ( Mallasto and Feragen,, 2017 ; Bachoc et al.,, 2017 ; Candelieri et al.,, 2022 )	$N$	$\mathcal{O}(N^{2}M^{2}d)$ to $\mathcal{O}(N^{2}M^{3})$
PWA-GP (separable Wasserstein)	$N$	$\mathcal{O}(N^{2}dM\log M)$
PCPWA (PCA-based PWA)	$N$	$\mathcal{O}(NMd^{2})$ (PCA) $+$ $\mathcal{O}(N^{2}kM\log M)$

Appendix

Table F.1: Leading-order cost of assembling the

N\times N

kernel matrix for the different methods, assuming

m_{i}\asymp M

samples per cloud. All methods then incur a shared

\mathcal{O}(N^{3})

cost for exact GP training.

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