标题： 用于降维的Kantorovich-Rubinstein距离 显示英文标题

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

摘要： 本论文的目标是研究Kantorovich-Rubinstein距离在构建分类问题中的样本复杂性描述子的应用。 其思路是利用Kantorovich-Rubinstein距离是在测度空间中的一种度量，同时考虑到底层度量空间的几何和拓扑结构。 我们为每个点类分配一个测度，并因此研究从这些测度之间的Kantorovich-Rubinstein距离中可以获得的几何信息。 我们证明，这些测度之间较大的Kantorovich-Rubinstein距离可以得出存在一个1-Lipschitz分类器能够良好分类点类的结论。 我们还讨论了Kantorovich-Rubinstein距离作为描述子的局限性。 显示英文摘要

摘要： 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.