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The Wasserstein distance provides a notion of dissimilarities betweenprobability measures, which has recent applications in learning of structureddata with varying size such as images and text documents. In this work, westudy the $k$-nearest neighbor classifier ($k$-NN) of probability measuresunder the Wasserstein distance. We show that the $k$-NN classifier is notuniversally consistent on the space of measures supported in $(0,1)$. As anyEuclidean ball contains a copy of $(0,1)$, one should not expect to obtainuniversal consistency without some restriction on the base metric space, or theWasserstein space itself. To this end, via the notion of $\sigma$-finite metricdimension, we show that the $k$-NN classifier is universally consistent onspaces 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 thespace of measures supported on a finite set, the space of Gaussian measures,and the space of measures with densities expressed as finite wavelet series.

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