A note on the richness of convex hulls of VC classes
Gàbor Lugosi, Pompeu Fabra University, Spain
Shahar Mendelson, The Australian National University, Australia
Vladimir Koltchinskii, The University of New Mexico, USA
Abstract
We prove the existence of a class A of subsets of
Rd 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
μ on Rd,
measurable set B and ε >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 ε. The question was motivated by the
investigation of the theoretical properties of certain algorithms
in machine learning.
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