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

TL;DR

This paper develops a $k$-means based method for estimating multiple trajectories from unlabeled data, demonstrating stable convergence with a rate of $n^{-1/2}$ as data size grows.

Contribution

It introduces a regularized variational approach for multi-trajectory estimation that leverages $k$-means, with proven convergence properties.

Findings

Estimators converge in Sobolev space with rate $n^{-1/2}$.

Method effectively addresses data association and trajectory estimation.

Approach is computationally simple and scalable.

Abstract

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 $n^{-1/2}$.