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

Convergence and Rates for Fixed-Interval Multiple-Track Smoothing Using $k$-Means Type Optimization