Separability of reproducing kernel spaces

TL;DR

This paper shows that certain function spaces with a Borel measurable feature map are guaranteed to be separable, which has implications for their analysis and application in machine learning.

Contribution

It establishes a new criterion linking Borel measurable feature maps to the separability of reproducing kernel spaces on complex spaces.

Findings

Reproducing kernel Hilbert or Banach spaces are separable if they have a Borel measurable feature map.

The result applies to spaces on separable absolute Borel or analytic subsets of Polish spaces.

Provides a theoretical foundation for analyzing the structure of kernel spaces in advanced settings.

Abstract

We demonstrate that a reproducing kernel Hilbert or Banach space of functions on a separable absolute Borel space or an analytic subset of a Polish space is separable if it possesses a Borel measurable feature map.