Operator-Valued Kernels, Machine Learning, and Dynamical Systems

TL;DR

This paper explores operator-valued kernels and their applications in machine learning and dynamical systems, providing new factorizations, realizations, and implications for Gaussian processes and non-commutative probability theory.

Contribution

It introduces new factorizations and realizations for positive operator-valued kernels and applies these results to Gaussian processes and non-commutative probability.

Findings

New factorizations of operator-valued kernels

Implications for Hilbert space-valued Gaussian processes

A novel non-commutative Radon--Nikodym theorem

Abstract

In the context of kernel optimization, we prove a result that yields new factorizations and realizations. Our initial context is that of general positive operator-valued kernels. We further present implications for Hilbert space-valued Gaussian processes, as they arise in applications to dynamics and to machine learning. Further applications are given in non-commutative probability theory, including a new non-commutative Radon--Nikodym theorem.