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The k-Nearest Neighbors (kNN) classifier is a fundamental non-parametricmachine learning algorithm. However, it is well known that it suffers from thecurse of dimensionality, which is why in practice one often applies a kNNclassifier on top of a (pre-trained) feature transformation. From a theoreticalperspective, most, if not all theoretical results aimed at understanding thekNN classifier are derived for the raw feature space. This leads to an emerginggap between our theoretical understanding of kNN and its practicalapplications. In this paper, we take a first step towards bridging this gap. Weprovide a novel analysis on the convergence rates of a kNN classifier overtransformed features. This analysis requires in-depth understanding of theproperties that connect both the transformed space and the raw feature space.More precisely, we build our convergence bound upon two key properties of thetransformed space: (1) safety -- how well can one recover the raw posteriorfrom the transformed space, and (2) smoothness -- how complex this recoveryfunction is. Based on our result, we are able to explain why some (pre-trained)feature transformations are better suited for a kNN classifier than other. Weempirically validate that both properties have an impact on the kNN convergenceon 30 feature transformations with 6 benchmark datasets spanning from thevision to the text domain.

On Convergence of Nearest Neighbor Classifiers over Feature Transformations