On improved bound for measure of cluster structure in compact metric spaces

TL;DR

This paper improves the theoretical bounds on the measure of balanced cluster structures in compact metric spaces by introducing a new restriction on distance distribution, leading to tighter asymptotic bounds.

Contribution

It introduces an additional restriction on distance distribution that enhances the bounds for the measure of maximal cluster structures in compact metric spaces.

Findings

New restriction improves asymptotic bounds

Bound for measure of cluster structures is significantly tighter

Results applicable to balanced clustering scenarios

Abstract

A compact metric space $(X, \rho)$ is given. Let $\mu$ be a Borel measure on $X$. By $r$-cluster we mean a measurable subset of $X$ with diameter at most $r$. A family of $k$ $2r$-clusters is called a $r$-cluster structure of order $k$ if any two clusters from the family are separated by a distance at least $r$. By measure of a cluster structure we mean a sum of clusters measures from the cluster structure. In our previous work we showed that under some parametric restrictions for distance distribution measure of maximal cluster structure $\mu(\mathcal{X})^*$ is close $\mu(X)$ and lower bound for $\mu(\mathcal{X})^*$ converges to $\mu(X)$ when corresponding parameters tend to 0. However, this bound asymptotically unimprovable. We propose an additional restriction for distance distribution that is responsible for balance of cluster's measure in cluster structure. This restriction allows to significantly improve previous bound in asymptotic sense.