Universal consistency of Wasserstein k-NN classifier
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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 analyze the k-nearest neighbor classifier (k-NN) under the Wasserstein distance and establish the universal consistency on families of distributions. Using previous known results on the consistency of the k-NN classifier on infinite dimensional metric spaces, it suffices to show that the families is a countable union of finite dimensional components. As a result, we are able to prove universal consistency of k-NN on spaces of finitely supported measures, the space of finite wavelet series and the spaces of Gaussian measures with commuting covariance matrices.
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