论文标题
关于随机采样的误差：参数曲线上均匀分布的随机点
On the Error of Random Sampling: Uniformly Distributed Random Points on Parametric Curves
论文作者
论文摘要
给定参数多项式曲线$γ：[a，b] \ rightarrow \ mathbb {r}^n $，我们如何以随机点$ \ mathfrak {x} \ in \ in \ mathrm {imrm {im}（γ）$进行均匀地与对ARC-LENGTER的均匀分布呢？不幸的是，我们不能完全采样这样的点 - 假设我们可以执行精确的算术操作。因此，我们最终提出了以下问题：我们选择的方法如何影响我们获得的近似样本的质量？实际上，有很多答案。但是，从理论上讲，我们的理解仍然存在差距。在本文中，我们从复杂性理论的角度解决了这个问题，从所需误差的大小方面提供了界限。
Given a parametric polynomial curve $γ:[a,b]\rightarrow \mathbb{R}^n$, how can we sample a random point $\mathfrak{x}\in \mathrm{im}(γ)$ in such a way that it is distributed uniformly with respect to the arc-length? Unfortunately, we cannot sample exactly such a point-even assuming we can perform exact arithmetic operations. So we end up with the following question: how does the method we choose affect the quality of the approximate sample we obtain? In practice, there are many answers. However, in theory, there are still gaps in our understanding. In this paper, we address this question from the point of view of complexity theory, providing bounds in terms of the size of the desired error.
