Universal consistency of Wasserstein k-NN classifier: a negative and some positive results

We study the $k$-nearest neighbour classifier ($k$-NN) of probability measures under the Wasserstein distance. We show that the $k$-NN classifier is not universally consistent on the space of measures supported in $(0,1)$. As any Euclidean ball contains a copy of $(0,1)$, one should not expect to obtain universal consistency without some restriction on the base metric space, or the Wasserstein space itself. To this end, via the notion of $\sigma $-finite metric dimension, we show that the $k$-NN classifier is universally consistent on spaces of discrete measures (and more generally, $\sigma $-finite uniformly discrete measures) with rational mass. In addition, by studying the geodesic structures of the Wasserstein spaces for $p=1$ and $p=2$, we show that the $k$-NN classifier is universally consistent on spaces of measures supported on a finite set, the space of Gaussian measures and spaces of measures with finite wavelet series densities.