Rate of Convergence for Smoothing Splines with Unknown Data Association

The problem of estimating multiple trajectories from unlabeled data comprises 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 variational problem which uses smoothing splines under a $k$-means like framework and show that, as the number of data points increases, we have stability. More precisely, we show that these estimators converge weakly in an appropriate Sobolev space with probability one. Furthermore, we show that the estimators converge in probability with rate $\frac{1}{\sqrt{n}}$ in the $L^2$ norm (strongly).