Product Kernel Function Overview

• Product kernel functions are defined as the product of component kernels, preserving positive definiteness over multivariate domains.
• They establish a tensor-product RKHS structure that enables efficient computation via Kronecker and Hadamard product algorithms.
• Their flexible framework supports applications in multivariate approximation, Gaussian processes, and feature decomposition in high-dimensional learning.

A product kernel function is a mathematically and computationally central construction in multivariate approximation, statistical learning theory, special function theory, and mathematical physics. It refers to a kernel on a product space (e.g., $\mathbb{R}^d$ or a product of domains or graph vertices) whose value at $(x,y)$ is the product of per-component or per-factor kernels acting on the corresponding marginals. Product kernels provide both theoretical and algorithmic advantages, including preservation of positive definiteness, native-space tensor-product structure, fast computation via Kronecker or Hadamard products, and natural boundary- or structure-awareness in domains with product geometry.

1. Formal Definition and Positive Definiteness

Given a product domain $$\Omega = \Omega^1 \times \ldots \times \Omega^M \subset \mathbb{R}^d$$, with $d = d_1 + \ldots + d_M$, and a collection of (positive semi-definite or positive definite) kernels $K_i: \Omega^i \times \Omega^i \to \mathbb{R}$ (typically $d_i=1$ in statistical learning, or $M=2$ for bipartite graphs or function spaces), the product kernel is defined by

$K(x, y) = \prod_{i=1}^M K_i(x^i, y^i)$

with $x = (x^1,\ldots,x^M)$ and $x^i \in \Omega^i$. In statistical machine learning, a common case is $K(x,y) = \prod_{j=1}^d k_j(x_j, y_j)$ for $x, y \in \mathbb{R}^d$.

If each $K_i$ is positive semi-definite (p.s.d.), then $K$ is p.s.d. by the Schur product theorem: the Gram matrix for a finite sample, $A_{K,X}$, is the Hadamard (entrywise) product of the component Gram matrices $A_{K_i, X^i}$ and thus is p.s.d. If each $K_i$ is strictly positive definite, then—under mild conditions—$K$ is strictly positive definite as well, both for arbitrary finite sets (by grid-embedding) and for sets of categorical or continuous grid-like structure (Albrecht et al., 2023).

2. Native Space Structure and Tensor Products

Every p.s.d. kernel $K$ on $\Omega$ defines a unique reproducing kernel Hilbert space (RKHS) $\mathcal{H}_{K,\Omega}$. For product kernels, the native space realizes a strict Hilbert tensor product structure: $$\mathcal{H}_{K,\Omega} \cong \mathcal{H}_{K_1, \Omega^1} \otimes_H \cdots \otimes_H \mathcal{H}_{K_M, \Omega^M}$$ with the inner product

$$\langle \varphi(f_1, \dots, f_M), \varphi(g_1, \dots, g_M) \rangle_K = \prod_{i=1}^M \langle f_i, g_i \rangle_{K_i}$$

and $$\varphi(f_1, \dots, f_M)(x) = \prod_{i=1}^M f_i(x^i)$$ (Albrecht et al., 2023). This tensor-product decomposition is central in approximation theory and in the analysis of Hardy spaces, weighted Bergman spaces, and their kernel functions over product domains (Guan et al., 2022).

3. Efficient Algorithms and Kronecker Structure

Product kernels are pivotal in enabling scalable computation in high-dimensional settings due to the Kronecker and Hadamard product algebraic structures:

• Grid-like sets: For $X = X^1 \times \ldots \times X^M$ (with $|X^i| = N_i$), the Gram matrix $A_{K,X}$ factors as a large Kronecker product: $A_{K,X} = \bigotimes_{i=1}^M A_{K_i, X^i}$. This structure is exploited for storage and arithmetic efficiency, e.g., reducing the cost of Cholesky decomposition and matrix solves in interpolation problems from $O((\prod N_i)^3)$ to $\sum O(N_i^3)$ (Albrecht et al., 2023).
• Kernel methods on graphs: In bipartite graphs or labeled edge prediction, the Gram matrix with a product kernel on edges is $R (G \otimes K) R^\top$, where $K$ and $G$ are Gram matrices on the vertex sets and $R$ is a permutation or selection operator. The so-called "vec-trick" or Roth's lemma allows rapid matrix-vector computations without explicitly instantiating the large $(mn) \times (mn)$ product (Airola et al., 2016).
• Gaussian processes: For product kernels, matrix-vector multiplication (MVM) can be performed with cost scaling only linearly with dimension (versus exponentially for generic SKI approaches), by low-rank factorization and recursive Hadamard product MVMs, enabling fully scalable GP regression in high dimensions (Gardner et al., 2018).

Table: Complexity Effect of Product vs. Generic Kernels in GPs

Kernel Structure	Memory	Per-MVM Cost
Generic SKI	$O(m^D)$	$O(n + m^D\log m)$
Product (SKIP)	$O(D m)$	$O(n + D m \log m)$

See (Gardner et al., 2018) for details.

4. Specialization: Product Kernels in Function and Special Function Theory

Product-type (or convolution-type) kernel identities are fundamental in the theory of special functions and integral transforms:

• Bessel functions: $$j_\alpha(u) j_\alpha(v) = C_\alpha \int_0^\pi j_\alpha(\sqrt{u^2 + v^2 - 2uv\cos\theta}) (\sin\theta)^{2\alpha} d\theta$$, representing the product as an integral over a kernel in yet another variable. Generalizations lead to product formulas for Hankel (Dunkl) transforms and enable the definition of translation and convolution operators with close analogy to the classical Fourier theory (Boubatra et al., 2020).
• Jacobi, Legendre, and Gegenbauer functions: Integral representations for products of Jacobi functions (of the second kind), Legendre functions, and their cut/Ferrers analogues all realize kernel representations analogous to product kernels, yielding Bateman-type expansions and Nicholson-type integral identities (Cohl et al., 11 Aug 2025).
• Hardy and Bergman spaces: On Hardy spaces over a product of planar domains, the kernel function for the space, under product weights, factors as a product of one-variable kernel functions. This factorization structure underpins results on extremality and comparison inequalities in the theory of spaces of holomorphic functions (Guan et al., 2022).

5. Interpretability and Feature Decomposition in Machine Learning

In statistical learning, the product-kernel structure enables exact functional ANOVA-type decompositions in RKHSs, where

$$k(x, x') = \prod_{d=1}^D k_d(x^{(d)}, x'^{(d)}) = \sum_{S \subset D} \prod_{d \in S} (k_d(x^{(d)}, x'^{(d)}) - 1)$$

leading to efficient computation of variable attributions (e.g., exact Shapley values) in time polynomial in $D$ (number of covariates), a task generically exponential in $D$. The PKeX-Shapley algorithm achieves $O(D^3)$ complexity for this computation when using product kernels, and the same principle extends to Shapley decompositions for statistical discrepancy measures (MMD, HSIC) (Mohammadi et al., 22 May 2025).

6. Domain-Adapted and Nonstationary Product Kernels

Beyond stationary settings, product kernels can be adapted to encode boundary or domain-specific priors. An example is the Beta-product kernel

$$K_\beta(x, x') = \prod_{i=1}^d \int_0^1 \mathrm{Beta}(s; \alpha_i, \beta_i) \mathrm{Beta}(s; \alpha'_i, \beta'_i) ds$$

with $x, x' \in [0,1]^d$, $\alpha_i = 1 + x_i/h_i$, $\beta_i = 1 + (1 - x_i)/h_i$, providing a nonstationary, boundary-aware kernel suitable for Bayesian optimization over bounded hypercubes. This kernel empirically exhibits exponential eigendecay comparable to the RBF, supporting efficient GP inference and superior performance when optima are near the boundaries of the domain (Nguyen et al., 19 Jun 2025).

7. Applications and Theoretical Extensions

Product kernels underpin a wide array of theoretical developments and practical algorithms:

• Multivariate interpolation and scattered data approximation, benefiting from Kronecker-structured Newton bases and improved scaling (Albrecht et al., 2023).
• Multi-output and multi-task learning, where product kernels encode task relations or domain decompositions (Airola et al., 2016, Gardner et al., 2018).
• Zero-shot generalization in graphs, enabling edge-level prediction for node pairs unseen in training (Airola et al., 2016).
• Bayesian optimization and model compression, with boundary-aware product kernels improving empirical regret and interpretability (Nguyen et al., 19 Jun 2025).
• Double Eisenstein or "kernel" constructions for products of $L$-values in analytic number theory, where the product structure realizes modular and Maass period identities (Diamantis et al., 2010).

Product kernels thus serve as a flexible, efficient, and theoretically robust framework, unifying disparate methodologies across computational mathematics, statistical machine learning, special function theory, and functional analysis. Their multiplicative structure delivers both practical speedups and deeper interpretability through functional and spectral decompositions.