Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Kernels, Distances, and Bridges

H. Movahedi-Lankarani, R. Wells

TL;DR

The paper broadens metric-space theory to general kernels $\kappa: X\times X\to\mathbb{R}$ by developing distances, almost distances, and the bridge construction on disjoint unions. It proves the existence of a largest below-kernel $\hat{\kappa}$ that is a distance, builds symmetric distances on unions via bridges, and extends these ideas to kernels on probability spaces with canonical $L^2(\mu)$ embeddings and associated Lipschitz structures. It also analyzes uniform point separation in metric-measure spaces and stability under perturbations, introducing a dilation-based pseudometric on bi-Lipschitz classes and showing how small changes preserve bi-Lipschitz embeddings. Together, these results unify kernel-based and metric-measure perspectives, providing tools for constructing and analyzing distances on unions and quotients and for understanding embeddings through operator- and measure-theoretic lenses.

Abstract

The purpose of this paper is to study more general real-valued functions of two variables than just metrics on a set X. We concentrate mainly on the classes of distances and almost distances. We also introduce the notion of a bridge on the disjoint union of two sets and show that it induces a symmetric distance on the disjoint union.

Kernels, Distances, and Bridges

TL;DR

The paper broadens metric-space theory to general kernels by developing distances, almost distances, and the bridge construction on disjoint unions. It proves the existence of a largest below-kernel that is a distance, builds symmetric distances on unions via bridges, and extends these ideas to kernels on probability spaces with canonical embeddings and associated Lipschitz structures. It also analyzes uniform point separation in metric-measure spaces and stability under perturbations, introducing a dilation-based pseudometric on bi-Lipschitz classes and showing how small changes preserve bi-Lipschitz embeddings. Together, these results unify kernel-based and metric-measure perspectives, providing tools for constructing and analyzing distances on unions and quotients and for understanding embeddings through operator- and measure-theoretic lenses.

Abstract

The purpose of this paper is to study more general real-valued functions of two variables than just metrics on a set X. We concentrate mainly on the classes of distances and almost distances. We also introduce the notion of a bridge on the disjoint union of two sets and show that it induces a symmetric distance on the disjoint union.

Paper Structure

This paper contains 7 sections, 24 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Distances
  3. Almost distances
  4. Bridges
  5. Kernels on a probability space
  6. More on uniform point separation
  7. Remarks

Key Result

Lemma 2.1

Let $\kappa \in \mathcal{K}^+ (X)$. Then there is a unique largest distance $\hat{\kappa} \le \kappa$. Moreover, $\hat{\kappa}$ is $\mathcal{T}_{\kappa} \times \mathcal{T}_{\kappa}$-continuous on $X_0 (\kappa) \times X_0 (\kappa)$.

Theorems & Definitions (27)

  • Lemma 2.1
  • proof
  • Corollary 2.2
  • Corollary 2.3
  • Lemma 2.4
  • Corollary 2.5
  • Corollary 2.6
  • Theorem 2.7: Inverse Function Theorem
  • proof
  • Lemma 3.1
  • ...and 17 more