On the minimum of the mean-squared error in 2-means clustering

TL;DR

This paper investigates the behavior of the minimum mean-squared error in 2-means clustering of two touching spheres in high-dimensional space, revealing dimension-dependent differences in optimal clustering strategies.

Contribution

It provides a detailed analysis of the asymptotic behavior of the minimum mean-squared error for 2-means clustering on touching spheres, highlighting the impact of dimension on cluster identification.

Findings

In dimensions n ≥ 3, the optimal partition separates the two spheres.

In dimension n=2, the optimal partition does not identify the individual spheres.

The study characterizes how the clustering error depends on the dimension and geometry.

Abstract

We study the minimum mean-squared error for 2-means clustering when the outcomes of the vector-valued random variable to be clustered are on two touching spheres of unit radius in $n$-dimensional Euclidean space and the underlying probability distribution is the normalized surface measure. For simplicity, we only consider the asymptotics of large sample sizes and replace empirical samples by the probability measure. The concrete question addressed here is whether a minimizer for the mean-squared error identifies the two individual spheres as clusters. Indeed, in dimensions $n \ge 3$, the minimum of the mean-squared error is achieved by a partition that separates the two spheres and has unit distance between the points in each cluster and the respective mean. In dimension $n=2$, however, the minimizer fails to identify the individual spheres; an optimal partition is obtained by a separating hyperplane that does not contain the point at which the spheres touch.