  We address the task of estimating multiple trajectories from unlabeled data.
This problem arises in many settings, one could think of the construction of
maps of transport networks from passive observation of travellers, or the
reconstruction of the behaviour of uncooperative vehicles from external
observations, for example. There are two coupled problems. The first is a data
association problem: how to map data points onto individual trajectories. The
second is, given a solution to the data association problem, to estimate those
trajectories. We construct estimators as a solution to a regularized
variational problem (to which approximate solutions can be obtained via the
simple, efficient and widespread $k$-means method) and show that, as the number
of data points, $n$, increases, these estimators exhibit stable behaviour. More
precisely, we show that they converge in an appropriate Sobolev space in
probability and with rate $n^{-1/2}$.
