Kriging via variably scaled kernels

Gianluca Audone; Francesco Marchetti; Emma Perracchione; Milvia Rossini

stat.MLMar 16, 2026

Kriging via variably scaled kernels

Authors: Gianluca Audone, Francesco Marchetti, Emma Perracchione, Milvia Rossini

Organizations: Faculty of Engineering and Physical Sciences, University of Southampton, United Kindom · Dipartimento di Matematica “Tullio Levi-Civita”, Universit`a di Padova, Italy · Dipartimento di Scienze Matematiche Giuseppe Luigi Lagrange, Politecnico di Torino, Italy · Universit`a degli Studi di Milano-Bicocca, Dipartimento di Matematica e Applicazioni, Italy

Abstract

Classical Gaussian processes and Kriging models are commonly based on stationary kernels, whereby correlations between observations depend exclusively on the relative distance between scattered data. While this assumption ensures analytical tractability, it limits the ability of Gaussian processes to represent heterogeneous correlation structures. In this work, we investigate variably scaled kernels as an effective tool for constructing non-stationary Gaussian processes by explicitly modifying the correlation structure of the data. Through a scaling function, variably scaled kernels alter the correlations between data and enable the modeling of targets exhibiting abrupt changes or discontinuities. We analyse the resulting predictive uncertainty via the variably scaled kernel power function and clarify the relationship between variably scaled kernels-based constructions and classical non-stationary kernels. Numerical experiments demonstrate that variably scaled kernels-based Gaussian processes yield improved reconstruction accuracy and provide uncertainty estimates that reflect the underlying structure of the data

Figures & tables

Figure 1: Stationary case (left) and VSK approach (right). The training data consist of

N=6

equispaced nodes.

Figure 2: Realization of the GPs using the prior (left) and posterior (right) covariance: stationary case (top) and VSK approach (bottom).

Panel of Figure 3 — Figure 3: Training matrices in the stationary (left) and VSK (right) case.

	Standard	VSK
RMSE	$9.1728\times 10^{-1}$	$\mathbf{8.4409\times 10^{-1}}$
Maximum error	$2.6058\times 10^{0}$	$\mathbf{2.3776\times 10^{0}}$
Average posterior std.	$\mathbf{2.3022\times 10^{-1}}$	$2.8207\times 10^{-1}$
Maximum posterior std.	$\mathbf{3.7459\times 10^{-1}}$	$8.7057\times 10^{-1}$

Table 1: Comparison between the stationary Matérn

C^{2}

kernel and the VSK kernel with fixed hyperparameters

\ell=1

\sigma_{f}=8

\sigma_{n}=0,

and

N=6

uniform nodes.

Figure 4: Approximation errors (left) and uncertainty (right) varying

N

from

10

790

Halton nodes.

Figure 5: Stationary case (left) and VSK approach (right). The training data consist of

N=27

Halton nodes.

Figure 6: Stationary case (left) and VSK approach (right). The training data consist of

N=81

Halton nodes.

Metric	Standard	VSK
Length scale $\ell$	0.0650	0.0627
Signal variance $\sigma_{f}$	1.4827	1.4956
Noise variance $\sigma_{n}$	0.3238	0.2428
RMSE	$5.3569\times 10^{-1}$	$3.6767\times 10^{-1}$
Max error	$3.0195\times 10^{0}$	$1.7015\times 10^{0}$
Average std	$4.6966\times 10^{-1}$	$4.0266\times 10^{-1}$

Table 2: Comparison between stationary kernel and VSK for

N=27

training Halton points (MLE-tuned hyperparameters).

Metric	Standard	VSK
Length scale $\ell$	0.0614	0.0694
Signal variance $\sigma_{f}$	1.6096	1.6613
Noise variance $\sigma_{n}$	0.3269	0.3321
RMSE	$2.4512\times 10^{-1}$	$1.3972\times 10^{-1}$
Max error	$2.0480\times 10^{0}$	$4.1026\times 10^{-1}$
Average std	$3.8423\times 10^{-1}$	$3.8782\times 10^{-1}$

Table 3: Comparison between stationary kernel and VSK for

N=81

training Halton points (MLE-tuned hyperparameters).

Figure 7: Approximation errors (left) and uncertainty (right) varying

K_{\rm VSK}

from

0

(stationary) to

12

Panel of Figure 8 — Figure 8: Approximation of the Weierstrass function, with confidence intervals. Top: the target function (left) and the stationary reconstruction (right). Bottom: VSK reconstructions with $K_{\rm VSK}=2$ (left), $K_{\rm VSK}=4$ (center) and $K_{\rm VSK}=12$ (right).

Figure 9: Approximation errors (left) and uncertainty (right) varying

N

from

11

791

Halton nodes.

Figure 10: Stationary case (left) and VSK approach (right). The training data consist of

N=21

uniform nodes.

Figure 11: Stationary case (left) and VSK approach (right). The training data consist of

N=20

uniform nodes.

Panel of Figure 12 — Figure 12: Training matrices in the stationary (left) and VSK (right) case: with $N=21$ (top) and $N=20$ (bottom) uniform nodes.

Figure 13: Stationary case (left), VSK approach (center) and a corresponding Gibbs kernel (right). The training data consist of

N=9

Chebyshev nodes.

Figure 14: Basis functions centered at

x^{\prime}=0

: stationary case (left), VSK approach (center) and a corresponding Gibbs kernel (right).

Gaussian Processes

Gaussian Processes are probabilistic models of functions that use kernels to make predictions with explicit uncertainty.

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Why not to use the Gaussian kernel

Toni Karvonen, Chris J. Oates

School of Engineering Sciences Lappeenranta–Lahti University of Technology LUT Yliopistonkatu 34, 53850 Lappeenranta, Finland · School of Mathematics, Statistics & Physics Newcastle University Newcastle upon Tyne, NE1 7RU, United Kingdom

Kernel Methods Gaussian Process Regression

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Kernels measure similarity or correlation in tasks such as regression and classification. The Gaussian kernel, other names of which include squared exponential and radial basis function kernel, is one of the most popular in Gaussian process regression. We argue that the Gaussian kernel is best avoided and should never be used as a default. The argument rests on two results demonstrating that the Gaussian kernel is extremely brittle. First, the Gaussian kernel gives rise to a conditional variance that is unrealistically small. If the variance is used to quantify predictive uncertainty, catastrophic overconfidence is almost inevitable. Second, a small variance goes hand in hand with numerical ill-conditioning, so that to use the Gaussian kernel in practice requires tricks such as nugget terms that effectively modify the underlying regression or classification model. These problems are caused by the unnatural smoothness of the Gaussian kernel, a fact we are far from the first to take notice of. The problem is not the Gaussian form itself but the analyticity of the kernel: Our argument is more broadly that analytic kernels are best avoided. For stationary kernels analyticity is essentially equivalent to an exponential decay of the spectral density.

Abstract

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