  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.
