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 dimension sets. As a result, we show that the $k$-NN classifier is universally consistent on spaces of finitely supported measures, the space of Gaussian measures, and the space of measures with finite wavelet densities. In addition, we give a counterexample to show that the universal consistency does not hold on $\mathcal{W}_p((0,1))$.