Subsampling in Smoothed Range Spaces

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 $\epsilon$-nets and $\epsilon$-samples (aka $\epsilon$-approximations). We characterize when size bounds for $\epsilon$-samples on kernels can be extended to these more general smoothed range spaces. We also describe new generalizations for $\epsilon$-nets to these range spaces and show when results from binary range spaces can carry over to these smoothed ones.

This is the full version of the paper which appeared in ALT 2015. 16 pages, 3 figures. In Algorithmic Learning Theory, pp. 224-238. Springer International Publishing, 2015

References in corpus (1)

• Epsilon-Nets for Halfspaces Revisited