In traditional Gaussian process (GP) regression, the inputs are typically assumed to be known, and uncertainty enters only through the measurements. When scalar fields are mapped using GP regression with measurements obtained from a mobile sensor, the inputs correspond to the estimated pose of the mobile sensor. The goal in this letter is to understand how a recorded GP, trained on erroneous inputs (e.g., pose estimates from dead-reckoning), can be corrected in a numerically efficient way when better estimates of the inputs become available. In the developed bias-aware GP (Ba-GP) framework, the recorded GP posterior-mean gradient is used to propagate estimated input bias and noise into mean and covariance corrections. The resulting additive covariance structure is then exploited to develop numerically efficient first-order correction algorithms. A simulation study and an indoor light-intensity mapping experiment show that Ba-GP improves predictive accuracy and uncertainty calibration relative to noisy-input GP baselines, with an order-of-magnitude reduction in computational cost relative to full GP retraining.
Figures & tables
Method
New matrix inverse
Cost
Full Ba-GP
Yes
O(N3)
Mean-only Ba-GP
No
O(N2)
Ba-GP Neumann (first order)
No
O(N2)
Ba-GP Neumann (second order)
No
O(N2)
TABLE I: Computational complexity of the Ba-GP variants.
Method
MAE ↓
NLPD ↓
Cov95
Reference GP
0.1230
−0.4519
0.9585
Recorded GP
0.2057
0.2514
0.8095
Retrained GP
0.1249
−0.4478
0.9720
Dallaire GP
0.2061
0.2426
0.8150
NIGP
0.2048
0.1785
0.8340
Offline Taylor series
0.1442
1.8860
0.9225
TABLE II: Simulation comparison across methods.
Fig. 1: Pairwise comparison of the eight GP regression methods against the reference GP. The dashed black curve denotes the known latent field h(x) , while the solid black curve denotes the reference GP trained at the exact input locations. Each colored curve denotes the corresponding method and the shaded region its ±2σ interval. The Retrained GP uses corrected training locations, whereas the recorded GP, Dallaire GP, NIGP, Full Ba-GP, Mean-only Ba-GP, and the first- and second-order Ba-GP Neumann approximations operate on the recorded inputs.
Fig. 2: Computational scaling of bias correction strategies with increasing N .
Fig. 3: Neumann convergence metric η as a function of input noise variance σx2 for different training-set sizes, while the dashed line denotes the Neumann-series convergence boundary η=1 .
Fig. 4: Relative Frobenius-norm error of the first- and second-order Neumann inverse approximations with respect to the exact Full Ba-GP inverse, plotted against the convergence metric η .
Fig. 5: Indoor light-mapping setup. The main image shows the ceiling-light arrangement, and the inset shows the Husarion ROSbot 2R Pro equipped with the Adafruit TSL2591 lux sensor used for data collection.
Fig. 6: Robot trajectory during light mapping showing the motion-capture reference sensing locations, aligned odometry sensing locations, and RTS-smoothed sensing locations.
Method
MAE ↓
NLPD ↓
Cov95
Reference GP
4.8864
3.3014
0.9799
Recorded GP
6.4609
3.5052
0.9436
Retrained GP
4.7189
3.2704
0.9843
Dallaire GP
6.4694
3.5023
0.9487
NIGP
6.4320
3.4912
0.9495
Full Ba-GP
4.7884
3.3071
0.9828
TABLE III: Experimental comparison across methods.
Fig. 7: Measured ground-level light-intensity field corresponding to the lighting layout in Fig. 5 .
Fig. 8: Normalized absolute prediction-error heat maps relative to the reference light-intensity field. The top panel shows the recorded GP prediction error with the odometry trajectory overlaid, while the bottom panel shows the Full Ba-GP prediction error with the RTS-smoothed trajectory overlaid.
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.
Hengrui Luo, Xiaoye S. Li, Yang Liu +3
Department of Statistics Rice University, Houston, TX 77005 · Lawrence Berkeley National Laboratory, Berkeley, CA 94709
Gaussian Processes (GPs) are a powerful tool for modelling and quantifying uncertainty in functional relationships. However, they require practitioners to make a number of design decisions, such as the choice of the kernel and the observation model. Suboptimal choices can produce misspecified models that do not capture the underlying data generating process. We introduce Predictively Oriented Gaussian Processes (PrO-GPs), which treat predictive uncertainty as the primary inferential target and provide a robust alternative to standard GPs. Although direct computation of a PrO posterior for nonparametric models is intractable, we derive a reduced formulation and practical sampling scheme for efficient computation. Through synthetic and real data experiments, we show that PrO-GPs produce better calibrated predictive distributions under model misspecification compared to standard GP approaches.
Nonstationary Gaussian process (GP) models are powerful tools for capturing input-dependent variability by adapting to observed data. However, with limited sampling and highly parameterized covariance structure, they are often prone to overfitting and overconfident uncertainty estimates, potentially leading to misleading predictions in safety-critical applications. Motivated by ionospheric modeling for satellite-based augmentation systems (SBAS), this paper proposes a Covariance-Boosted Gaussian Process (CBGP) framework centered upon boosting covariance priors to discover nonstationary latent functions for signal and observation variation that capture irregularities in the input domain. An additional layer of GP modeling of "partially-whitened" observations guides latent function relative error estimation that is used to iteratively update weak priors in a gradient descent-like procedure. Following boosting, restrictions are imposed upon prior covariances to prevent overfitting while posterior uncertainties are inflated to prevent model overconfidence. CBGP model efficacy and robustness are demonstrated through out-of-sample testing of both simulated and real-world applications that meet a three-nines integrity standard. The modeling of an extensive ionospheric storm dataset over South America suggests accurate and reliable means to compute SBAS ionospheric corrections in the most challenging space weather environment using regional models that are more informed and responsive than local fitting performed by currently-operating SBAS.
Jeremy Ovadia
Sequoia Research Corporation 2875 Temple Ave Signal Hill, CA, 90755, USA