An approach to the Gaussian RBF kernels via Fock spaces

TL;DR

This paper explores the Gaussian RBF kernel using Fock space and Segal-Bargmann theories, revealing connections with quantum mechanics operators and providing new mathematical insights relevant to machine learning.

Contribution

It introduces a novel approach to analyze Gaussian RBF kernels through complex analysis, linking them to quantum operators and the Segal-Bargmann transform.

Findings

Connections between RBF kernels and quantum operators established

Semigroup property of Weyl operators studied

Feature space and map characterized via complex analysis

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 machines (SVMs) classification algorithms. Complex analysis techniques allow us to consider several notions linked to the RBF kernels like the feature space and the feature map, using the so-called Segal-Bargmann transform. We show also 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.