On the Error of Random Sampling: Uniformly Distributed Random Points on Parametric Curves

TL;DR

This paper investigates the theoretical limitations and complexity bounds of uniformly sampling points on parametric polynomial curves with respect to arc-length, highlighting the inherent challenges and errors involved.

Contribution

It provides the first complexity bounds for approximate uniform sampling on parametric curves, bridging a gap in theoretical understanding.

Findings

Establishes bounds on sampling error based on complexity theory.

Shows exact uniform sampling is impossible with finite computation.

Highlights the impact of method choice on sample quality.

Abstract

Given a parametric polynomial curve $\gamma:[a,b]\rightarrow \mathbb{R}^n$, how can we sample a random point $\mathfrak{x}\in \mathrm{im}(\gamma)$ 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.