  Building upon recent advances in entropy-regularized optimal transport, and
upon Fenchel duality between measures and continuous functions , we propose a
generalization of the logistic loss that incorporates a metric or cost between
classes. Unlike previous attempts to use optimal transport distances for
learning, our loss results in unconstrained convex objective functions,
supports infinite (or very large) class spaces, and naturally defines a
geometric generalization of the softmax operator. The geometric properties of
this loss make it suitable for predicting sparse and singular distributions,
for instance supported on curves or hyper-surfaces. We study the theoretical
properties of our loss and show-case its effectiveness on two applications:
ordinal regression and drawing generation.
