On Clusters that are Separated but Large

TL;DR

This paper investigates the problem of partitioning a set of points into large, well-separated clusters in high-dimensional space, providing tight bounds and algorithms based on parameters like separation, spread, and number of clusters.

Contribution

It offers tight bounds and algorithms for clustering points into large, well-separated groups considering parameters like separation, spread, and the number of clusters.

Findings

Derived tight upper and lower bounds on cluster separation and size.

Developed algorithms achieving these bounds under given parameters.

Analyzed the influence of spread and separation on clustering quality.

Abstract

$\renewcommand{\Re}{\mathbb{R}}$Given a set $P$ of $n$ points in $\Re^d$, consider the problem of computing $k$ subsets of $P$ that form clusters that are well-separated from each other, and each of them is large (cardinality wise). We provide tight upper and lower bounds, and corresponding algorithms, on the quality of separation, and the size of the clusters that can be computed, as a function of $n,d,k,s$, and $\Phi$, where $s$ is the desired separation, and $\Phi$ is the spread of the point set $P$.