Supervised learning of sheared distributions using linearized optimal transport

TL;DR

This paper introduces a method for supervised learning on probability measures by embedding them into $L^2$ spaces via optimal transport, enabling linear separation of classes, including sheared distributions, with applications to image classification.

Contribution

It extends optimal transport embedding techniques to include sheared distributions and provides conditions and bounds for linear separability in this context.

Findings

Successful linear separation of sheared distributions.

Derived bounds for transformation parameters to achieve separation.

Demonstrated effectiveness on image classification tasks.

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 $L^2$ 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.