On the Separability of Stochastic Geometric Objects, with Applications

TL;DR

This paper develops efficient algorithms to compute the probability of linear separability and the expected separation margin for stochastic geometric objects, extending to polytopes and balls, with applications to stochastic convex hull problems.

Contribution

It introduces optimal algorithms for separability probability and margin in stochastic geometric models, including extensions to general objects and complexity bounds.

Findings

Separable probability computed in O(nN^{d-1}) for d≥3

Expected separation margin computed in O(nN^{d}) for d≥2

Algorithms extended to polytopes and balls with polynomial time complexity

Abstract

In this paper, we study the linear separability problem for stochastic geometric objects under the well-known unipoint/multipoint uncertainty models. Let $S=S_R \cup S_B$ be a given set of stochastic bichromatic points, and define $n = \min\{|S_R|, |S_B|\}$ and $N = \max\{|S_R|, |S_B|\}$. We show that the separable-probability (SP) of $S$ can be computed in $O(nN^{d-1})$ time for $d \geq 3$ and $O(\min\{nN \log N, N^2\})$ time for $d=2$, while the expected separation-margin (ESM) of $S$ can be computed in $O(nN^{d})$ time for $d \geq 2$. In addition, we give an $\Omega(nN^{d-1})$ witness-based lower bound for computing SP, which implies the optimality of our algorithm among all those in this category. Also, a hardness result for computing ESM is given to show the difficulty of further improving our algorithm. As an extension, we generalize the same problems from points to general geometric objects, i.e., polytopes and/or balls, and extend our algorithms to solve the generalized SP and ESM problems in $O(nN^{d})$ and $O(nN^{d+1})$ time, respectively. Finally, we present some applications of our algorithms to stochastic convex-hull related problems.