Probabilistic Embedding Of Discrete Sets As Continuous Metric Spaces

TL;DR

This paper explores how symmetric affinity functions on discrete sets induce Euclidean structures, enabling visualization of probabilistic loci for various geometric configurations.

Contribution

It introduces a method to represent discrete sets with affinity functions as continuous metric spaces and visualizes probabilistic loci for specific geometric structures.

Findings

Visual representations of probabilistic loci for a chain, polyhedron, and 2D lattice.

Demonstrates the Euclidean space structure induced by affinity functions.

Shows how graph-based affinity matrices relate to metric topological spaces.

Abstract

Any symmetric affinity function $w: V\times V \to \mathbb{R}_+$ defined on a discrete set $V$ induces Euclidean space structure on $V$. In particular, an undirected graph specified by an affinity (or adjacency) matrix can be considered as a metric topological space. We have calculated the visual representations of the probabilistic locus for a chain, a polyhedron, and a finite 2-dimensional lattice.