K-Means-Type Algorithms: A Generalized Convergence Theorem and Characterization of Local Optimality

TLDR

K‑means is a widely used clustering method. The paper investigates key questions about the K‑means algorithm. The authors formulate clustering as a nonconvex program and propose a method to find local‑minimum solutions. They prove finite convergence for any metric, demonstrate that convergence to a local minimum can fail under certain conditions, and show that differentiability ensures convergence to a Kuhn‑Tucker point.

Abstract

The K-means algorithm is a commonly used technique in cluster analysis. In this paper, several questions about the algorithm are addressed. The clustering problem is first cast as a nonconvex mathematical program. Then, a rigorous proof of the finite convergence of the K-means-type algorithm is given for any metric. It is shown that under certain conditions the algorithm may fail to converge to a local minimum, and that it converges under differentiability conditions to a Kuhn-Tucker point. Finally, a method for obtaining a local-minimum solution is given.