Persistent Clustering and a Theorem of J. Kleinberg

TL;DR

This paper introduces a new framework for clustering that incorporates persistence and functoriality, leading to a unique, stable clustering scheme with proven properties, contrasting Kleinberg's non-existence theorem.

Contribution

It develops a novel clustering framework with multiresolution and comparability features, proving a unique existence theorem and analyzing stability and convergence.

Findings

Existence and uniqueness of a clustering scheme within the framework

The scheme is stable under data perturbations

The scheme converges with increasing data size

Abstract

We construct a framework for studying clustering algorithms, which includes two key ideas: persistence and functoriality. The first encodes the idea that the output of a clustering scheme should carry a multiresolution structure, the second the idea that one should be able to compare the results of clustering algorithms as one varies the data set, for example by adding points or by applying functions to it. We show that within this framework, one can prove a theorem analogous to one of J. Kleinberg, in which one obtains an existence and uniqueness theorem instead of a non-existence result. We explore further properties of this unique scheme, stability and convergence are established.