  The k-Nearest Neighbors (kNN) classifier is a fundamental non-parametric
machine learning algorithm. However, it is well known that it suffers from the
curse of dimensionality, which is why in practice one often applies a kNN
classifier on top of a (pre-trained) feature transformation. From a theoretical
perspective, most, if not all theoretical results aimed at understanding the
kNN classifier are derived for the raw feature space. This leads to an emerging
gap between our theoretical understanding of kNN and its practical
applications. In this paper, we take a first step towards bridging this gap. We
provide a novel analysis on the convergence rates of a kNN classifier over
transformed features. This analysis requires in-depth understanding of the
properties that connect both the transformed space and the raw feature space.
More precisely, we build our convergence bound upon two key properties of the
transformed space: (1) safety -- how well can one recover the raw posterior
from the transformed space, and (2) smoothness -- how complex this recovery
function 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. We
empirically validate that both properties have an impact on the kNN convergence
on 30 feature transformations with 6 benchmark datasets spanning from the
vision to the text domain.
