Worst-Case and Smoothed Analysis of the Hartigan-Wong Method for k-Means Clustering

TL;DR

This paper investigates the Hartigan-Wong method for k-means clustering, revealing exponential worst-case running time but demonstrating that smoothed analysis shows polynomial bounds on expected iterations after Gaussian perturbations.

Contribution

It provides the first exponential worst-case example for the Hartigan-Wong method and establishes polynomial smoothed bounds on its expected running time under Gaussian noise.

Findings

Exponential worst-case running time demonstrated.

Polynomial expected running time under smoothed analysis.

Smoothed analysis framework applied to k-means clustering algorithm.

Abstract

We analyze the running time of the Hartigan-Wong method, an old algorithm for the $k$-means clustering problem. First, we construct an instance on the line on which the method can take $2^{\Omega(n)}$ steps to converge, demonstrating that the Hartigan-Wong method has exponential worst-case running time even when $k$-means is easy to solve. As this is in contrast to the empirical performance of the algorithm, we also analyze the running time in the framework of smoothed analysis. In particular, given an instance of $n$ points in $d$ dimensions, we prove that the expected number of iterations needed for the Hartigan-Wong method to terminate is bounded by $k^{12kd}\cdot poly(n, k, d, 1/\sigma)$ when the points in the instance are perturbed by independent $d$-dimensional Gaussian random variables of mean $0$ and standard deviation $\sigma$.