On the Use of the Kantorovich-Rubinstein Distance for Dimensionality
  Reduction
Gaël Giordano
The goal of this thesis is to study the use of the Kantorovich-Rubinstein
distance as to build a descriptor of sample complexity in classification
problems. The idea is to use the fact that the Kantorovich-Rubinstein distance
is a metric in the space of measures that also takes into account the geometry
and topology of the underlying metric space. We associate to each class of
points a measure and thus study the geometrical information that we can obtain
from the Kantorovich-Rubinstein distance between those measures. We show that a
large Kantorovich-Rubinstein distance between those measures allows to conclude
that there exists a 1-Lipschitz classifier that classifies well the classes of
points. We also discuss the limitation of the Kantorovich-Rubinstein distance
as a descriptor.