Sample Path Regularity of Gaussian Processes from the Covariance Kernel
CoRR(2023)
摘要
Gaussian processes (GPs) are the most common formalism for defining
probability distributions over spaces of functions. While applications of GPs
are myriad, a comprehensive understanding of GP sample paths, i.e. the function
spaces over which they define a probability measure on, is lacking. In
practice, GPs are not constructed through a probability measure, but instead
through a mean function and a covariance kernel. In this paper we provide
necessary and sufficient conditions on the covariance kernel for the sample
paths of the corresponding GP to attain a given regularity. We use the
framework of H\"older regularity as it grants us particularly straightforward
conditions, which simplify further in the cases of stationary and isotropic
GPs. We then demonstrate that our results allow for novel and unusually tight
characterisations of the sample path regularities of the GPs commonly used in
machine learning applications, such as the Mat\'ern GPs.
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