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

Donlapark Ponnoprat

Abstract: 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 σ-finite metric dimension, we show that the k-NN classifier is universally consistent on spaces of discrete measures (and more generally, σ-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.