Simplified and Improved Bounds on the VC-Dimension for Elastic Distance Measures

Abstract

We study range spaces, where the ground set consists of either polygonal curves in $\mathbb{R}^d$ or polygonal regions in the plane that may contain holes and the ranges are balls defined by an elastic distance measure, such as the Hausdorff distance, the Fr\'echet distance and the dynamic time warping distance. The range spaces appear in various applications like classification, range counting, density estimation and clustering when the instances are trajectories, time series or polygons. The Vapnik-Chervonenkis dimension (VC-dimension) plays an important role when designing algorithms for these range spaces. We show for the Fr\'echet distance of polygonal curves and the Hausdorff distance of polygonal curves and planar polygonal regions that the VC-dimension is upper-bounded by $O(dk\log(km))$ where $k$ is the complexity of the center of a ball, $m$ is the complexity of the polygonal curve or region in the ground set, and $d$ is the ambient dimension. For $d \geq 4$ this bound is tight in each of the parameters $d, k$ and $m$ separately. For the dynamic time warping distance of polygonal curves, our analysis directly yields an upper-bound of $O(\min(dk^2\log(m),dkm\log(k)))$.