  In the context of kernel methods, the similarity between data points is
encoded by the kernel function which is often defined thanks to the Euclidean
distance, a common example being the squared exponential kernel. Recently,
other distances relying on optimal transport theory - such as the Wasserstein
distance between probability distributions - have shown their practical
relevance for different machine learning techniques. In this paper, we study
the use of exponential kernels defined thanks to the regularized Wasserstein
distance and discuss their positive definiteness. More specifically, we define
Wasserstein feature maps and illustrate their interest for supervised learning
problems involving shapes and images. Empirically, Wasserstein squared
exponential kernels are shown to yield smaller classification errors on small
training sets of shapes, compared to analogous classifiers using Euclidean
distances.
