Universal consistency of Wasserstein $k$-NN classifier

The Wasserstein distance provides a notion of dissimilarities between probability measures, which has recent applications in learning of structured data with varying size such as images and text documents. In this work, we analyze the $k$-nearest neighbor classifier ($k$-NN) under the Wasserstein distance and establish the universal consistency on families of distributions. Using previous known results on the consistency of the $k$-NN classifier on infinite dimensional metric spaces, it suffices to show that the families is a countable union of finite dimensional components. As a result, we are able to prove universal consistency of $k$-NN on spaces of finitely supported measures, the space of finite wavelet series and the spaces of Gaussian measures with commuting covariance matrices.