Wasserstein Exponential Kernels

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; the squared exponential kernel is a common example. 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.