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2003 A note on the richness of convex hulls of VC classes
Gábor Lugosi, Shahar Mendelson, Vladimir Koltchinskii
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Electron. Commun. Probab. 8: 167-169 (2003). DOI: 10.1214/ECP.v8-1097

Abstract

We prove the existence of a class $A$ of subsets of $\mathbb{R}^d$ of VC dimension 1 such that the symmetric convex hull $F$ of the class of characteristic functions of sets in $A$ is rich in the following sense. For any absolutely continuous probability measure $\mu$ on $\mathbb{R}^d$, measurable set $B$ and $\varepsilon \gt 0$, there exists a function $f$ in $F$ such that the measure of the symmetric difference of $B$ and the set where $f$ is positive is less than $\varepsilon$. The question was motivated by the investigation of the theoretical properties of certain algorithms in machine learning.

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Gábor Lugosi. Shahar Mendelson. Vladimir Koltchinskii. "A note on the richness of convex hulls of VC classes." Electron. Commun. Probab. 8 167 - 169, 2003. https://doi.org/10.1214/ECP.v8-1097

Information

Accepted: 17 December 2003; Published: 2003
First available in Project Euclid: 18 May 2016

zbMATH: 1095.28006
MathSciNet: MR2042755
Digital Object Identifier: 10.1214/ECP.v8-1097

Subjects:
Primary: 62G08, 68Q32

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