On the Use of the Kantorovich-Rubinstein Distance for Dimensionality Reduction

TL;DR

This paper explores using the Kantorovich-Rubinstein distance to measure sample complexity in classification, leveraging its geometric properties to inform classifier effectiveness and limitations.

Contribution

It introduces a novel approach to assess sample complexity in classification using the Kantorovich-Rubinstein distance, linking geometric measure differences to classifier performance.

Findings

Large Kantorovich-Rubinstein distance indicates existence of effective 1-Lipschitz classifiers.

The distance captures geometric and topological information relevant to classification.

Limitations of the distance as a descriptor are discussed.

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.