Unified Native Spaces in Kernel Methods

TL;DR

This work introduces a unified, parametrized class of radial kernels, $\mathcal{H}_{\boldsymbol{\theta}}$, that encompasses many classical kernels by exact parameterizations or asymptotics. It establishes the spectral and Sobolev-analytic structure of this class, proving that the associated RKHS is norm-equivalent to the Sobolev space $H^{\alpha-k}(\mathbb{R}^d)$ under precise conditions and that the kernels can exhibit either compact or global support as well as hole effects. The framework recovers notable kernels such as Matérn and Wendland as special or limiting cases and provides explicit turning bands constructions and analytical representations in terms of hypergeometric and Meijer-$G$ functions. These results enable precise control over smoothness, local regularity, and correlation features, with implications for kriging, SPDE approaches, Bayesian nonparametrics, and large-scale kernel methods across statistics, ML, and numerical analysis.

Abstract

There exists a plethora of parametric models for positive definite kernels, and their use is ubiquitous in disciplines as diverse as statistics, machine learning, numerical analysis, and approximation theory. Usually, the kernel parameters index certain features of an associated process. Amongst those features, smoothness (in the sense of Sobolev spaces, mean square differentiability, and fractal dimensions), compact or global supports, and negative dependencies (hole effects) are of interest to several theoretical and applied disciplines. This paper unifies a wealth of well-known kernels into a single parametric class that encompasses them as special cases, attained either by exact parameterization or through parametric asymptotics. We furthermore characterize the Sobolev space that is norm equivalent to the RKHS associated with the new kernel. As a by-product, we infer the Sobolev spaces that are associated with existing classes of kernels. We illustrate the main properties of the new class, show how this class can switch from compact to global supports, and provide special cases for which the kernel attains negative values over nontrivial intervals. Hence, the proposed class of kernel is the reproducing kernel of a very rich Hilbert space that contains many special cases, including the celebrated Matérn and Wendland kernels, as well as their aliases with hole effects.

Figures (5)

• Figure 1: Connections between covariance kernels. Blue boxes are compactly-supported kernels; yellow boxes are hole effect kernels; green boxes are compactly-supported hole effect kernels. Solid arrows indicate particular cases; dashed arrows indicate asymptotic cases. Connections established in previous literature are indicated in blue; connections proved in this paper are indicated in black.
• Figure 2: Hole effect truncated polynomial kernel ${\cal H}_{\boldsymbol{\theta}}$ for $\boldsymbol{\theta}=(a,\frac{d+1}{2}+k+p,1+\frac{d+1}{2}+k+p+M,1+\frac{d}{2}+k+N,d,k)^{\top}$, for different choices of $k$, $p$, $M$ and $N$.
• Figure 3: Hole effect Askey ${\cal W}_{a,0,\nu,d,k}$ (left) and ordinary Wendland ${\cal W}_{a,1,\nu,d,k}$ (right) kernels, for $\nu = 6$, $d=2$ and $k = 0, 1, 2$.
• Figure 4: ${\cal W}_{\mu a,\nu-\frac{1}{2},\mu,d,k}$ and ${\cal M}_{a,\nu,d,k}$ (red line) when $\mu=10, 50$, $\nu = 1.5$, $d=2$ and $k=0$ (left) or $k=2$ (right).
• Figure 5: ${\cal M}_{a,k+\frac{1}{2},d,k}$ when $d=2$ and $k=5, 10, 100$ and ${\cal J}_{a,d}$ (green line).