Supervised learning of sheared distributions using linearized optimal transport
This work addresses classification tasks involving probability measures, such as in image processing, but is incremental as it extends an existing framework to include shearing transformations.
The paper tackles the problem of supervised learning on probability measures by embedding them into L^2 spaces using optimal transport, extending the framework to handle sheared distributions and providing conditions for linear separability. It demonstrates results on image classification tasks with necessary bounds on transformations to achieve specified separation levels.
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.