On Kernels and Covariance Structures in Hilbert Space Gaussian Processes
Saeed Hashemi Sababe
TL;DR
This work extends reproducing-kernel Hilbert-space (RKHS) theory to operator-valued kernels $K:S × S → B(H)$ and builds Hilbert-space–valued Gaussian processes with well-defined covariance structures. It constructs the RKHS $H_{ ilde{K}}$ from the induced scalar kernel $\tilde{K}((s,a),(t,b))=\langle a, K(s,t)b\rangle_H$ and introduces a feature-map family $\{V_s\}$ with $K(s,t)=V_s^*V_t$ and covariance $\Sigma_s=V_s^*V_s$, enabling a clear dilation/embedding interpretation. The main results establish strong continuity of $V_s$ over topological domains, extend to vector- and matrix-valued kernels, and connect covariance operators to kernel-valued mappings through identities like $\Sigma_s=V_s^*V_s$ and $V_sV_s^*$ acting as covariance operators in $H_{\tilde{K}}$; additional extensions include operator transforms $B_s$ and compactness properties of the induced operators. These contributions provide a rigorous foundation for analyzing infinite-dimensional covariance structures in operator-valued settings, with potential impact on quantum computing, functional data analysis, and high-dimensional stochastic modeling.
Abstract
Motivated by practical applications, I present a novel and comprehensive framework for operator-valued positive definite kernels. This framework is applied to both operator theory and stochastic processes. The first application focuses on various dilation constructions within operator theory, while the second pertains to broad classes of stochastic processes. In this context, the authors utilize the results derived from operator-valued kernels to develop new Hilbert space-valued Gaussian processes and to investigate the structures of their covariance configurations.