On the tightness of an SDP relaxation of k-means

TL;DR

This paper analyzes the conditions under which a semidefinite programming relaxation accurately recovers clusters in a probabilistic model of data points distributed around fixed centers, showing high-probability exact recovery when centers are sufficiently separated.

Contribution

It provides a theoretical analysis of the tightness of an SDP relaxation for k-means in a random data model, establishing explicit separation conditions for exact recovery.

Findings

Exact recovery with high probability when centers are separated by more than 2 + epsilon.

Recovery probability approaches 1 exponentially fast in the number of points n.

Separation epsilon can be made arbitrarily small in high-dimensional settings.

Abstract

Recently, Awasthi et al. introduced an SDP relaxation of the $k$-means problem in $\mathbb R^m$. In this work, we consider a random model for the data points in which $k$ balls of unit radius are deterministically distributed throughout $\mathbb R^m$, and then in each ball, $n$ points are drawn according to a common rotationally invariant probability distribution. For any fixed ball configuration and probability distribution, we prove that the SDP relaxation of the $k$-means problem exactly recovers these planted clusters with probability $1-e^{-\Omega(n)}$ provided the distance between any two of the ball centers is $>2+\epsilon$, where $\epsilon$ is an explicit function of the configuration of the ball centers, and can be arbitrarily small when $m$ is large.