Convergence of the $k$-Means Minimization Problem using $\Gamma$-Convergence

TL;DR

This paper extends the $k$-means algorithm to infinite-dimensional problems using $\Gamma$-convergence, showing convergence of minima and minimizers, and demonstrates applications in data association and sensor tracking.

Contribution

It introduces a $\Gamma$-convergence framework for $k$-means, enabling its application to infinite-dimensional problems and complex data scenarios.

Findings

Convergence of $k$-means minima and minimizers in large data limit.

Application to non-parametric smoothing with unknown data association.

Application to sparse sensor network tracking.

Abstract

The $k$-means method is an iterative clustering algorithm which associates each observation with one of $k$ clusters. It traditionally employs cluster centers in the same space as the observed data. By relaxing this requirement, it is possible to apply the $k$-means method to infinite dimensional problems, for example multiple target tracking and smoothing problems in the presence of unknown data association. Via a $\Gamma$-convergence argument, the associated optimization problem is shown to converge in the sense that both the $k$-means minimum and minimizers converge in the large data limit to quantities which depend upon the observed data only through its distribution. The theory is supplemented with two examples to demonstrate the range of problems now accessible by the $k$-means method. The first example combines a non-parametric smoothing problem with unknown data association. The second addresses tracking using sparse data from a network of passive sensors.