The ratio-cut polytope and K-means clustering

TL;DR

This work considers the stochastic ball model, a popular generative model for K-means clustering, and shows that if the separation distance between cluster centers satisfies $\Delta > 1+\sqrt 3$, then the LP relaxation recovers the planted clusters with high probability.

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 $Δ> 1+\sqrt 3$, 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 $Δ> 4$. Our numerical experiments indicate that the proposed LP relaxation significantly outperforms a popular semidefinite programming relaxation in recovering the planted clusters.

Figures (5)

• Figure 1: The polytopes ${\rm RCut}^{K}_n$ and ${\rm RCut}^{=K}_n$, with $n =3$ and $K=2$. The ratio-cut polytope ${\rm RCut}^{K}_n$ is the convex hull of the ratio-cut vectors $\left\{(\frac{1}{3},\frac{1}{3},\frac{1}{3}), (\frac{1}{2},0,0), (0,\frac{1}{2},0), (0,0,\frac{1}{2})\right\}$. The polytope ${\rm RCut}^{=K}_n$, shaded in red, is a facet of ${\rm RCut}^{K}_n$ whose affine hall is given by $X_{12} + X_{13} + X_{23} = \frac{1}{2}$.
• Figure 2: The empirical probability of success of the LP versus the SDP in recovering the planted clusters when the points are generated according to the SBM.
• Figure 3: Cluster centers $\gamma^k, k\in [K]$ with hive-shaped geometry. The parameter $\Delta$ is defined as the distance between two adjacent centers.
• Figure 4: The empirical probability of success of the weakest LP versus the SDP in recovering planted clusters for the SBM in dimension two, with $K =4$ clusters (Figure \ref{['fig3a']}) and with $K =5$ clusters (Figure \ref{['fig3b']}).
• Figure 5: Comparing the quality of two LP relaxations, LPt2 vs LPt3 for the SBM in dimension two, with $K =4$ clusters (Figure \ref{['fig4a']}) and with $K =5$ clusters (Figure \ref{['fig4b']}).

Theorems & Definitions (27)

• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Proposition 4
• proof
• Proposition 5
• proof