Convergence of the k-Means Minimization Problem Using Gamma-Convergence

Matthew Thorpe, Florian Theil, Adam Johansen, Neil Cade

Research output: Contribution to journalArticlepeer-review


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 Γ-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.
Original languageEnglish
Pages (from-to)2444-2474
Number of pages31
JournalSIAM Journal of Applied Mathematics
Issue number6
Publication statusPublished - 2015


  • Clustering
  • smoothing
  • variational approach


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