A notion of stability for k-means clustering

TL;DR

This paper introduces a new stability concept for k-means clustering based on the quantization of probability measures, linking it to a geometric property called the absolute margin condition.

Contribution

It defines a novel stability notion for k-means and connects it to the geometric absolute margin condition of data distributions.

Findings

Establishes a relationship between stability and the absolute margin condition.

Provides theoretical insights into the geometric properties influencing k-means stability.

Abstract

In this paper, we define and study a new notion of stability for the $k$-means clustering scheme building upon the notion of quantization of a probability measure. We connect this notion of stability to a geometric feature of the underlying distribution of the data, named absolute margin condition, inspired by recent works on the subject.