k-NN Regression Adapts to Local Intrinsic Dimension
published: Jan. 25, 2012, recorded: December 2011, views: 5472
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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.
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