29 research outputs found
Convergence and Rates for Fixed-Interval Multiple-Track Smoothing Using -Means Type Optimization
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 -means method) and show that, as the number
of data points, , increases, these estimators exhibit stable behaviour. More
precisely, we show that they converge in an appropriate Sobolev space in
probability and with rate
On smoothed analysis of quicksort and Hoare's find
We provide a smoothed analysis of Hoare's find algorithm, and we revisit the smoothed analysis of quicksort. Hoare's find algorithm - often called quickselect or one-sided quicksort - is an easy-to-implement algorithm for finding the k-th smallest element of a sequence. While the worst-case number of comparisons that Hoareās find needs is Theta(n^2), the average-case number is Theta(n). We analyze what happens between these two extremes by providing a smoothed analysis. In the first perturbation model, an adversary specifies a sequence of n numbers of [0,1], and then, to each number of the sequence, we add a random number drawn independently from the interval [0,d]. We prove that Hoare's find needs Theta(n/(d+1) sqrt(n/d) + n) comparisons in expectation if the adversary may also specify the target element (even after seeing the perturbed sequence) and slightly fewer comparisons for finding the median. In the second perturbation model, each element is marked with a probability of p, and then a random permutation is applied to the marked elements. We prove that the expected number of comparisons to find the median is Omega((1āp)n/p log n). Finally, we provide lower bounds for the smoothed number of comparisons of quicksort and Hoareās find for the median-of-three pivot rule, which usually yields faster algorithms than always selecting the first element: The pivot is the median of the first, middle, and last element of the sequence. We show that median-of-three does not yield a significant improvement over the classic rule