Using Existential Theory of the Reals to Bound VC Dimension
Using Existential Theory of the Reals to Bound VC Dimension
复制标题
DOI：
--
复制
发表时间：
2022
期刊：
影响因子：
--
通讯作者：
Austin Watkins
中科院分区：
文献类型：
--
作者：
Austin Watkins
我们在多项式和其他离散的几何形状的逻辑组成之外提供了范围空间的VC尺寸，我们的结果解决了一个看似简单的范围空间的VC维度，我们称为跨度的多项式，这些空间被定义为Minkowski的Minkowski总和多项式和r 2中的球为θ（p），在r d中，绑定为o（dp o（d））关于多项式经典的框架和多项式轨迹的对抗性的问题，我们使用代数几何形状和基于经典电路的方法来界定VC维度来得出我们的结果，我们相信我们的总体结果和我们的总体结果。可以在学习理论，范围搜索和计算几何形状的其他方面中找到其他应用，其中VC维度起着关键作用。
We provide new bounds on the VC dimension of range spaces beyond logical compositions of polynomials and other discrete geometric shapes. Our results address the VC dimension of a seemingly simple class of range spaces we call inflated polynomials, which are defined as the Minkowski sum of a polynomial and a ball; in R 2 with degree p the VC dimension is Θ( p ) , and in R d the bound is O ( dp O ( d ) ) . This addresses natural questions on learnability in the adversarially-robust setting for polynomial classifiers and of polynomially-defined trajectories. We use a connection between algebraic geometry and classic circuit-based approaches of bounding the VC dimension to derive our results. We believe this connection and our general results may find other applications in learning theory, range searching, and other aspects of computational geometry where the VC dimension plays a key role.