Geometry of a Set and its Random covers

TL;DR

This paper investigates the probability that random balls centered at uniform samples from a set cover the set, deriving bounds based on geometric properties and partitions, with applications in approximation and coverage analysis.

Contribution

It introduces geometric conditions and partitioning methods, including Whitney decomposition and multiscale flat norm, to establish coverage probability bounds for random ball unions.

Findings

Lower bounds tend to 1 as exp(-δ^n N)

Good partitions exist if the complement has positive reach

Multiscale flat norm helps approximate sets without positive reach

Abstract

Let $E$ be a bounded open subset of $\mathbb{R}^n$. We study the following questions: For i.i.d. samples $X_1, \dots, X_N$ drawn uniformly from $E$, what is the probability that $\cup_i \mathbf{B}(X_i, \delta)$, the union of $\delta$-balls centered at $X_i$, covers $E$? And how does the probability depend on sample size $N$ and the radius of balls $\delta$? We present geometric conditions of $E$ under which we derive lower bounds to this probability. These lower bounds tend to $1$ as a function of $\exp{(-\delta^n N)}$. The basic tool that we use to derive the lower bounds is a good partition of $E$, i.e., one whose partition elements have diameters that are uniformly bounded from above and have volumes that are uniformly bounded from below. We show that if $E^c$, the complement of $E$, has positive reach then we can construct a good partition of $E$. This partition is motivated by the Whitney decomposition of $E$. On the other hand, we identify a class of bounded open subsets of $\mathbb{R}^n$ that do not satisfy this positive reach condition but do have good partitions. In 2D when $E^c\subset \mathbb{R}^2$ does not have positive reach, we show that the mutliscale flat norm can be used to approximate $E$ with a set that has a good partition under certain conditions. In this case, we provide a lower bound on the probability that the union of the balls almost covers $E$.