A Gentle Introduction to the Kernel Distance

TL;DR

This paper introduces the kernel distance, explaining its interpretation as an L2 distance in an RKHS, and discusses its applications and mathematical foundations for data analysis involving probability measures and geometric shapes.

Contribution

It provides a gentle, accessible introduction to the kernel distance, emphasizing its interpretation and applications in data analysis and geometric measure theory.

Findings

Kernel distance as L2 measure in RKHS

Efficient solutions for data analysis problems

Mathematical foundations linking to multiple fields

Abstract

This document reviews the definition of the kernel distance, providing a gentle introduction tailored to a reader with background in theoretical computer science, but limited exposure to technology more common to machine learning, functional analysis and geometric measure theory. The key aspect of the kernel distance developed here is its interpretation as an L_2 distance between probability measures or various shapes (e.g. point sets, curves, surfaces) embedded in a vector space (specifically an RKHS). This structure enables several elegant and efficient solutions to data analysis problems. We conclude with a glimpse into the mathematical underpinnings of this measure, highlighting its recent independent evolution in two separate fields.