# Subsampling in Smoothed Range Spaces

@inproceedings{Phillips2015SubsamplingIS,
title={Subsampling in Smoothed Range Spaces},
author={J. M. Phillips and Yan Zheng},
booktitle={ALT},
year={2015}
}
• Published in ALT 4 October 2015
• Computer Science, Mathematics
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…
2 Citations
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• 2021
A method for subsampling of spatial data suitable for creating kernel density estimates from very large data is described and it is demonstrated that it results in less error than random sampling.
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• Computer Science
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• 2017
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## References

SHOWING 1-10 OF 44 REFERENCES
Є-Samples for Kernels
It turns out, the use of this smoother family of range spaces has an added benefit of greatly decreasing the size required for e-samples, for instance, in the plane the size is greatly decreasing.
Comparing distributions and shapes using the kernel distance
• Computer Science
SoCG '11
• 2011
This paper presents fast approximation algorithms for computing the kernel distance between two point sets P and Q that runs in near-linear time in the size of P ∪ Q (an explicit calculation would take quadratic time).
Small-size ε-nets for axis-parallel rectangles and boxes
• Mathematics, Computer Science
STOC '09
• 2009
Improved approximation factors are obtained for the hitting set or the set cover problems associated with the corresponding range spaces for ε-nets of size O(1/ε log log log 1/ε) for planar point sets and axis-parallel rectangular ranges.
Small-Size $\eps$-Nets for Axis-Parallel Rectangles and Boxes
• Computer Science, Mathematics
SIAM J. Comput.
• 2010
Improved approximation factors for the hitting set or the set cover problems associated with the corresponding range spaces are obtained by plugging the bounds into the technique of Bronnimann and Goodrich or of Even, Rawitz, and Shahar.
Algorithms for ε-approximations of Terrains ?
Consider a point set D with a measure function μ : D→ R. Let A be the set of subsets of D induced by containment in a shape from some geometric family (e.g. axis-aligned rectangles, half planes,
Tight lower bounds for the size of epsilon-nets
• Mathematics, Computer Science
SoCG '11
• 2011
It is shown that there exist geometrically defined range spaces, already of VC-dimension 2, in which the size of the smallest ε-nets is Ω(1/ε log 1/ε).
On Range Searching in the Group Model and Combinatorial Discrepancy
• Kasper Green Larsen
• Mathematics, Computer Science
2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
• 2011
An intimate connection between dynamic range searching in the group model and combinatorial discrepancy is established and a whole range of exceptionally high and near-tight lower bounds for all of the basic range searching problems are established.
Scale-sensitive dimensions, uniform convergence, and learnability
• Mathematics
JACM
• 1997
A characterization of learnability in the probabilistic concept model, solving an open problem posed by Kearns and Schapire, and shows that the accuracy parameter plays a crucial role in determining the effective complexity of the learner's hypothesis class.
PiCoDes: Learning a Compact Code for Novel-Category Recognition
• Computer Science
NIPS
• 2011
PICODES: a very compact image descriptor which nevertheless allows high performance on object category recognition and an alternation scheme and convex upper bound which demonstrate excellent performance in practice are presented.
New existence proofs ε-nets
• Mathematics
SCG '08
• 2008
A new technique is described for proving the existence of small μ-nets for hypergraphs satisfying certain simple conditions and is particularly useful for proving o(1/μ log 1/μ) upper bounds which the standard VC-dimension theory does not allow.