An approach to the Gaussian RBF kernels via Fock spaces

Abstract

We use methods from the Fock space and Segal–Bargmann theories to prove several results on the Gaussian RBF kernel in complex analysis. The latter is one of the most used kernels in modern machine learning kernel methods and in support vector machine classification algorithms. Complex analysis techniques allow us to consider several notions linked to the radial basis function (RBF) kernels, such as the feature space and the feature map, using the so-called Segal–Bargmann transform. We also show how the RBF kernels can be related to some of the most used operators in quantum mechanics and time frequency analysis; specifically, we prove the connections of such kernels with creation, annihilation, Fourier, translation, modulation, and Weyl operators. For the Weyl operators, we also study a semigroup property in this case.