Universal consistency of Wasserstein $k$-NN classifier: Negative and Positive Results

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 study the $k$-nearest neighbor 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 measures supported in a $\sigma$-uniformly discrete set. 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 the space of measures supported on a finite set, the space of Gaussian measures, and the space of measures with densities expressed as finite wavelet series.