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k-NN Regression Adapts to Local Intrinsic Dimension

2011

Conference Paper

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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.

Author(s): Kpotufe, S.
Book Title: Advances in Neural Information Processing Systems 24
Pages: 729-737
Year: 2011
Day: 0
Editors: J Shawe-Taylor and RS Zemel and P Bartlett and F Pereira and KQ Weinberger

Department(s): Empirical Inference
Bibtex Type: Conference Paper (inproceedings)

Event Name: Twenty-Fifth Annual Conference on Neural Information Processing Systems (NIPS 2011)
Event Place: Granada, Spain

Digital: 0

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BibTex

@inproceedings{Kpotufe2012,
  title = {k-NN Regression Adapts to Local Intrinsic Dimension},
  author = {Kpotufe, S.},
  booktitle = {Advances in Neural Information Processing Systems 24},
  pages = {729-737},
  editors = {J Shawe-Taylor and RS Zemel and P Bartlett and F Pereira and KQ Weinberger},
  year = {2011}
}