The ratio-cut polytope and K-means clustering

TL;DR

This paper introduces the ratio-cut polytope and a new LP relaxation for K-means clustering, providing theoretical guarantees for exact recovery under certain conditions and demonstrating superior empirical performance.

Contribution

It defines the ratio-cut polytope, analyzes its facial structure, and develops a novel LP relaxation for K-means with improved recovery guarantees under the stochastic ball model.

Findings

LP relaxation recovers clusters if separation > 1+√3

Outperforms SDP relaxation in experiments

Provides new facet inequalities for the ratio-cut polytope

Abstract

We introduce the ratio-cut polytope defined as the convex hull of ratio-cut vectors corresponding to all partitions of $n$ points in $\mathbb R^m$ into at most $K$ clusters. This polytope is closely related to the convex hull of the feasible region of a number of clustering problems such as K-means clustering and spectral clustering. We study the facial structure of the ratio-cut polytope and derive several types of facet-defining inequalities. We then consider the problem of K-means clustering and introduce a novel linear programming (LP) relaxation for it. Subsequently, we focus on the case of two clusters and derive sufficient conditions under which the proposed LP relaxation recovers the underlying clusters exactly. Namely, we consider the stochastic ball model, a popular generative model for K-means clustering, and we show that if the separation distance between cluster centers satisfies $\Delta > 1+\sqrt 3$, then the LP relaxation recovers the planted clusters with high probability. This is a major improvement over the only existing recovery guarantee for an LP relaxation of K-means clustering stating that recovery is possible with high probability if and only if $\Delta > 4$. Our numerical experiments indicate that the proposed LP relaxation significantly outperforms a popular semidefinite programming relaxation in recovering the planted clusters.