用$k$ -均值型Optimization进行固定间隔多道平滑的收敛性和速度
Convergence and Rates for Fixed-Interval Multiple-Track Smoothing Using $k$-Means Type Optimization
摘要：
我们解决的任务，估计多个轨迹从未标记的数据。这个问题出现在许多环境中，例如，人们可以想到通过对旅行者的被动观察来构建交通网络的地图，或者通过外部观察来重建不合作车辆的行为。有两个相互关联的问题。第一个是数据关联问题：如何将数据点映射到单个轨迹上。第二个是，给定数据关联问题的解决方案，以估计这些轨迹。我们构造估计的正则化变分问题的解决方案（通过简单，高效和广泛的$k$ -手段方法可以得到近似解），并表明，作为数据点的数量，$n$，增加，这些估计表现出稳定的行为。更精确地说，我们证明了它们在一个适当的索伯列夫空间中以概率和速率$n^{-1/2}$收敛。
英文摘要：
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 $n^{-1/2}$.
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