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In this paper we study supervised learning tasks on the space of probabilitymeasures. We approach this problem by embedding the space of probabilitymeasures into $L^2$ spaces using the optimal transport framework. In theembedding spaces, regular machine learning techniques are used to achievelinear separability. This idea has proved successful in applications and whenthe classes to be separated are generated by shifts and scalings of a fixedmeasure. This paper extends the class of elementary transformations suitablefor the framework to families of shearings, describing conditions under whichtwo classes of sheared distributions can be linearly separated. We furthermoregive necessary bounds on the transformations to achieve a pre-specifiedseparation level, and show how multiple embeddings can be used to allow forlarger families of transformations. We demonstrate our results on imageclassification tasks.

Supervised learning of sheared distributions using linearized optimal transport