  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.
