Gaussian Processes Generated By Monotonically Modulated Stationary Kernels

TL;DR

This paper studies Gaussian processes created by modulating stationary kernels monotonically, establishing an isometry between their Hilbert spaces and deriving an exact formula for the expected number of zeros based on kernel derivatives and the modulation function.

Contribution

It introduces a novel class of Gaussian processes with monotonic kernel modulation and provides explicit mathematical relationships and properties, including eigenvalue preservation and zero count formulas.

Findings

Isometry between original and modulated RKHSs preserving eigenvalues

Exact expression for the expected number of zeros of the process

Relationship between kernel derivatives and zero count

Abstract

This article examines Gaussian processes generated by monotonically modulating stationary kernels. An explicit isometry between the original and the modulated reproducing kernel Hilbert spaces is established, preserving eigenvalues and normalization. The expected number of zeros over the interval $[0,T]$ is shown to be exactly $\sqrt{-\ddot{K}(0)}(\theta(T)-\theta(0))$, where $\ddot{K}(0)$ is the second derivative of the kernel at zero and $\theta$ is the modulating function.