New boundaries for positive definite functions

TL;DR

This paper introduces new theoretical frameworks connecting positive definite kernels with Gaussian processes, geometric boundaries, and applications in machine learning, expanding understanding of kernel factorizations and their diverse applications.

Contribution

It develops novel correspondences and factorizations for positive definite kernels, linking probabilistic and geometric perspectives, with applications to boundary analysis and machine learning.

Findings

New boundary analysis for the Drury-Arveson kernel

Identification of optimal feature spaces in machine learning

Factorizations linking kernels to Gaussian processes and geometry

Abstract

With view to applications in stochastic analysis and geometry, we introduce a new correspondence for positive definite kernels (p.d.) $K$ and their associated reproducing kernel Hilbert spaces. With this we establish two kinds of factorizations: (i) Probabilistic: Starting with a positive definite kernel $K$ we analyze associated Gaussian processes $V$. Properties of the Gaussian processes will be derived from certain factorizations of $K$, arising as a covariance kernel of $V$. (ii) Geometric analysis: We discuss families of measure spaces arising as boundaries for $K$. Our results entail an analysis of a partial order on families of p.d. kernels, a duality for operators and frames, optimization, Karhunen--Lo\`eve expansions, and factorizations. Applications include a new boundary analysis for the Drury-Arveson kernel, and for certain fractals arising as iterated function systems; and an identification of optimal feature spaces in machine learning models.