The Vapnik–Chervonenkis dimension provides a notion of complexity for systems of sets. If the VC dimension is small, then knowing this can drastically simplify fundamental computational tasks such as classification, range counting, and density estimation through the use of sampling bounds. We analyze set systems where the ground set X is a set of polygonal curves in \(\mathbb {R}^d\) and the sets \(\mathcal {R}\) are metric balls defined by curve similarity metrics, such as the Fréchet distance and the Hausdorff distance, as well as their discrete counterparts. We derive upper and lower bounds on the VC dimension that imply useful sampling bounds in the setting that the number of curves is large, but the complexity of the individual curves is small. Our upper and lower bounds are either near-quadratic or near-linear in the complexity of the curves that define the ranges and they are logarithmic in the complexity of the curves that define the ground set.

Vapnik-Chervonenkis 维为集合系统提供了复杂性的概念。如果 VC 维度很小，那么知道这一点可以通过使用采样边界大大简化基本的计算任务，例如分类、范围计数和密度估计。我们分析集合系统，其中地面集合X是\(\mathbb {R}^d\)和集合\(\mathcal {R}\) 中的一组多边形曲线是由曲线相似性度量定义的度量球，例如 Fréchet 距离和 Hausdorff 距离，以及它们的离散对应项。我们推导出 VC 维度的上限和下限，这意味着在曲线数量很大但单个曲线的复杂度很小的情况下有用的采样边界。我们的上限和下限在定义范围的曲线的复杂性上接近二次或接近线性，并且在定义地面集的曲线的复杂性上是对数的。