On the Use of the Kantorovich-Rubinstein Distance for Dimensionality Reduction
Abstract
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.
Keywords
Cite
@article{arxiv.2309.09442,
title = {On the Use of the Kantorovich-Rubinstein Distance for Dimensionality Reduction},
author = {Gaël Giordano},
journal= {arXiv preprint arXiv:2309.09442},
year = {2023}
}
Comments
214 pages, 0 figures, This is a PhD thesis in mathematics under the supervision of Dr. Vladimir Pestov and Dr. George Wells submitted on May 1, 2023 at the University of Ottawa