Fréchet和Hausdorff距离下度量球的VC维数
The VC Dimension of Metric Balls under Fréchet and Hausdorff Distances
摘要：
Vapnik-Chervonenkis维数为集合系统提供了复杂性的概念。如果VC维很小，那么知道这一点可以大大简化基本的计算任务，如分类，范围计数和密度估计，通过使用采样边界。我们分析集系统的地面集$X$是一组多边形曲线在$\mathbb{R}^d$和集$\mathcal{R}$是度量球定义的曲线相似性度量，如Fréchet距离和Hausdorff距离，以及它们的离散对应。我们推导出VC维上的上界和下界，这意味着在曲线的数量很大的设置中有用的采样界，但单个曲线的复杂性很小。我们的上界在定义范围的曲线的复杂性上是近二次的或近线性的，并且在定义基集的曲线的复杂性上是对数的。
英文摘要：
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 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.
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