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. [ABSTRACT FROM AUTHOR]