Covariance Regularity Index (CRI)

• CRI is a numerical invariant assigned to positive-definite kernels that precisely measures the maximal Hölder-like regularity of Gaussian process sample paths.
• It provides necessary and sufficient analytic criteria based on mixed derivative and local increment conditions to ensure desired smoothness.
• Its applications include guiding kernel selection in Bayesian modeling, spatial statistics, and other machine learning methods that require controlled sample path regularity.

The Covariance Regularity Index (CRI) is a numerical invariant assigned to any positive-definite kernel that captures the maximal order of Hölder-like path regularity exhibited almost surely by sample paths of the Gaussian process (GP) it induces. CRI serves as a precise quantitative bridge between the analytic structure of covariance kernels and the stochastic regularity properties of GPs, providing necessary and sufficient conditions on the kernel for a desired sample-path regularity order. This concept allows unified, tight characterizations of the regularity of sample paths, particularly in machine learning applications using stationary and isotropic kernels, including the widely used Matérn, Wendland, and squared-exponential families (Costa et al., 2023).

1. Formal Definition

Let $O\subset\mathbb{R}^d$ be open and $k:O\times O\to\mathbb{R}$ a positive-definite kernel. The Covariance Regularity Index of $k$ is defined as

$\mathrm{CRI}(k) := \sup\big\{s\geq 0:\ \text{$k$satisfies the conditions of Theorem 1 with$n=\lfloor s\rfloor$and$\gamma=s-\lfloor s\rfloor$}\big\}.$

Equivalently,

$$\mathrm{CRI}(k)=\sup \big\{s\geq 0:\ f\sim \mathrm{GP}(0,k) \text{ a.s.\ } f\in C^{(s)^-}_{\mathrm{loc}}(O)\big\},$$

where $C^{(s)^-}_{\mathrm{loc}}(O)$, the local almost-Hölder space of order $s$, is defined as the intersection of $C^{n',\gamma'}_{\mathrm{loc}}(O)$ spaces for all $n'+\gamma'<n+\gamma$ if $s=n+\gamma$, $n\in\mathbb{N}_0$, $\gamma\in(0,1]$. Here, $C^{n,\gamma}_{\mathrm{loc}}(O)$ denotes functions with derivatives up to order $n$ whose $n$th partial derivatives are locally $\gamma$-Hölder.

2. Kernel-Derivative Conditions Characterizing CRI

For $s=n+\gamma$, $n=\lfloor s\rfloor \in \mathbb{N}_0$, $\gamma\in(0,1]$, the kernel $k$ satisfies $\mathrm{CRI}(k)\geq s$ if and only if:

• (a) Regularity: $k\in C^{n\otimes n}(O\times O)$, i.e., all mixed derivatives $\partial_x^\alpha \partial_y^\beta k(x,y)$ exist and are continuous for multi-indices $|\alpha|,|\beta|\leq n$.
• (b) Increment Condition: For every compact $K\subset O$, there exists $C_K<\infty$ such that for all $x\in K$ and small $h\in\mathbb{R}^d$,

$$\big|\partial^{\alpha,\beta}k(x+h,x+h) - \partial^{\alpha,\beta}k(x+h,x) - \partial^{\alpha,\beta}k(x,x+h) + \partial^{\alpha,\beta}k(x,x)\big| \leq C_K\|h\|^{2\gamma}$$

whenever $|\alpha|=|\beta|=n$.

For stationary kernels $k(x,y)=k_\delta(x-y)$, the conditions reduce to $k_\delta\in C^{2n}(\mathbb{R}^d)$ and $$|\partial^\alpha k_\delta(h)-\partial^\alpha k_\delta(0)|=O(\|h\|^{2\gamma})$$ for all $|\alpha|=2n$ as $h\to 0$. For isotropic kernels $k(x,y)=k_r(\|x-y\|)$, the requirements become $k_r\in C^{2n}(\mathbb{R})$ and $|k_r^{(2n)}(h)-k_r^{(2n)}(0)|= O(|h|^{2\gamma})$ as $h\to 0$.

3. Relationship Between CRI and Sample Path Hölder Exponent

By construction,

$$\mathrm{CRI}(k) = \sup\{s\geq 0: f\sim \mathrm{GP}(0,k) \text{ a.s. has sample paths in } C^{(s)^-}_{\mathrm{loc}}(O)\}.$$

This implies that if $\mathrm{CRI}(k)=s$, then almost surely $f\in C^{(s)^-}$, but $f\notin C^{(s')^-}$ for any $s'>s$. The almost sure local Hölder exponent $\alpha$ of the process is given by $\alpha = \mathrm{CRI}(k)$. This construction provides a sharp measure of the maximal path regularity compatible with the underlying covariance structure.

4. CRI of Standard Kernel Families

Explicit calculation of CRI values for key stationary and isotropic kernels yields:

Kernel Family	Regularity Parameter	CRI Value
Matérn ($k_r(r)$)	smoothness $\nu>0$	CRI$(k)=\nu$
Wendland ($k_r(r)$)	degree $n\in\mathbb{N}_0$	CRI$(k) = n+\tfrac12$
Squared-Exponential, Rational-Quadratic, Periodic	n/a	CRI$(k)=\infty$

• Matérn kernel: With $$k_r(r) = 2^{1-\nu}/\Gamma(\nu) \cdot (\sqrt{2\nu}r)^{\nu} K_\nu(\sqrt{2\nu}r)$$, it holds that $k_r\in C^{2\lfloor\nu\rfloor}$ and $$|k_r^{(2\lfloor\nu\rfloor)}(h) - k_r^{(2\lfloor\nu\rfloor)}(0)| = O(|h|^{2(\nu-\lfloor\nu\rfloor)})$$, hence $\mathrm{CRI}(k)=\nu$. For half-integer $\nu=n+\frac12$, $\mathrm{CRI}(k)=n+\frac12$; sample paths are exactly $n$-times continuously differentiable.
• Wendland kernel: Piecewise polynomial with degree depending on $d$ and $n$, the first non-smooth term appearing at order $2n+1$. $2n$-th derivative is Lipschitz, so $\mathrm{CRI}(k)=n+\frac12$.
• Infinitely smooth kernels: Such as squared-exponential, rational-quadratic, periodic, have $k_r\in C^\infty$, thus $\mathrm{CRI}(k)=\infty$, and the GP sample paths are almost surely $C^\infty$.

5. Simplification in Special and Composite Cases

• If $k\in C^\infty(O\times O)$ globally, then $\mathrm{CRI}(k)=\infty$.
• For feature map kernels, $k(x,y)=\phi(x)\cdot\phi(y)$ with $\phi: O\to\mathbb{R}^m$, one has $$\mathrm{CRI}(k)=\min_i \mathrm{CRI}(\text{feature } \phi_i)$$; for linear and polynomial cases, $\mathrm{CRI}=\infty$.
• For kernel sums and products as well as coordinate transformations,
  • $$\mathrm{CRI}(k\oplus k') = \min\{\mathrm{CRI}(k),\mathrm{CRI}(k')\}$$.
  • For coordinate transforms $\phi$, $\mathrm{CRI}(k\circ\phi)=\mathrm{CRI}(k)\cdot$(regularity of $\phi$).

These results are derived by tracking the behavior of mixed derivatives and their local increments under these operations.

6. Significance and Applications

The CRI provides a universal regularity index summarizing the maximal achievable Hölder-type regularity of GP sample paths determined solely by the local behavior of the covariance kernel around the diagonal—or, in translation-invariant forms, around the origin. By giving necessary and sufficient analytic criteria, CRI streamlines the design and selection of kernels for applications where sample-path regularity is critical, such as Bayesian function-space modeling, spatial statistics, and scientific machine learning. The index is especially useful in comparing and classifying kernels by the stochastic smoothness properties they induce.

7. Further Implications

A direct implication is that the CRI is uniquely determined by the analytic order of Hölder-like differentiability of $k$ around the diagonal, precisely equating to the almost-sure Hölder exponent of $f\sim\mathrm{GP}(0,k)$. This enables exact control over GP regularity via kernel choice, and underpins tight characterizations of path spaces for commonly used families in Gaussian process modeling (Costa et al., 2023).