Supervised learning of sheared distributions using linearized optimal transport

Abstract

In this paper we study supervised learning tasks on the space of probability measures. We approach this problem by embedding the space of probability measures into $$L^2$$ L 2 spaces using the optimal transport framework. In the embedding spaces, regular machine learning techniques are used to achieve linear separability. This idea has proved successful in applications and when the classes to be separated are generated by shifts and scalings of a fixed measure. This paper extends the class of elementary transformations suitable for the framework to families of shearings, describing conditions under which two classes of sheared distributions can be linearly separated. We furthermore give necessary bounds on the transformations to achieve a pre-specified separation level, and show how multiple embeddings can be used to allow for larger families of transformations. We demonstrate our results on image classification tasks.

Statistical learning on measures: an application to persistence diagrams

“…The theory for simple kernel estimators has been developed in Szabó et al (2016). Another recent perspective regarding distribution learning follows the works by Moosmüller and Cloninger (2020) and Khurana et al (2023), where the authors consider that each class consists of perturbations of a "mother distribution" and tackle this problem using tools from optimal transport. To conclude our overview of measure-learning methods, we can cite the work from , where the authors vectorize the measures to cluster them or perform a supervised learning task.…”