TY - JOUR

T1 - Convergence and Rates for Fixed-Interval Multiple-Track Smoothing Using $k$-Means Type Optimization

AU - Thorpe, Matthew

AU - Johansen, Adam

PY - 2016

Y1 - 2016

N2 - 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}.

AB - 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}.

U2 - https://projecteuclid.org/euclid.ejs/1480734075

DO - https://projecteuclid.org/euclid.ejs/1480734075

M3 - Article

SN - 1935-7524

VL - 10

SP - 3693

EP - 3722

JO - Electronic Journal of Statistics

JF - Electronic Journal of Statistics

IS - 2

ER -