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k-NN Regression Adapts to Local Intrinsic Dimension

MPG-Autoren
http://pubman.mpdl.mpg.de/cone/persons/resource/persons84024

Kpotufe,  S
Department Empirical Inference, Max Planck Institute for Biological Cybernetics, Max Planck Society;

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Zitation

Kpotufe, S. (2012). k-NN Regression Adapts to Local Intrinsic Dimension. In Advances in Neural Information Processing Systems 24 (pp. 729-737). Red Hook, NY, USA: Curran.


Zitierlink: http://hdl.handle.net/11858/00-001M-0000-0013-B87C-1
Zusammenfassung
Many nonparametric regressors were recently shown to converge at rates that depend only on the intrinsic dimension of data. These regressors thus escape the curse of dimension when high-dimensional data has low intrinsic dimension (e.g. a manifold). We show that k-NN regression is also adaptive to intrinsic dimension. In particular our rates are local to a query x and depend only on the way masses of balls centered at x vary with radius. Furthermore, we show a simple way to choose k = k(x) locally at any x so as to nearly achieve the minimax rate at x in terms of the unknown intrinsic dimension in the vicinity of x. We also establish that the minimax rate does not depend on a particular choice of metric space or distribution, but rather that this minimax rate holds for any metric space and doubling measure.