Subsampling in Smoothed Range Spaces

TL;DR

This paper explores smoothed geometric range spaces where elements have non-binary containment values, extending kernel-based notions to general ranges, and investigates how approximation techniques like ε-nets and ε-samples apply to these spaces.

Contribution

It characterizes conditions under which size bounds for ε-samples in kernels extend to smoothed range spaces and introduces new generalizations for ε-nets in this context.

Findings

Extended kernel ε-sample bounds to smoothed range spaces

Developed new ε-net generalizations for smoothed spaces

Identified conditions for transferring binary range space results

Abstract

We consider smoothed versions of geometric range spaces, so an element of the ground set (e.g. a point) can be contained in a range with a non-binary value in $[0,1]$. Similar notions have been considered for kernels; we extend them to more general types of ranges. We then consider approximations of these range spaces through $\varepsilon $-nets and $\varepsilon $-samples (aka $\varepsilon$-approximations). We characterize when size bounds for $\varepsilon $-samples on kernels can be extended to these more general smoothed range spaces. We also describe new generalizations for $\varepsilon $-nets to these range spaces and show when results from binary range spaces can carry over to these smoothed ones.