Rates of convergence of the functional k-nearest neighbor estimate

Authors:
Gérard Biau;Frédéric Cérou;Arnaud Guyader
Affiliations:
LSTA and LPMA, Université Pierre et Marie Curie-Paris VI, Paris, France;INRIA Rennes, Rennes Cedex, France;INRIA Rennes, Rennes Cedex, France and Université Rennes 2, Haute Bretagne, Rennes Cedex, France
Venue:
IEEE Transactions on Information Theory
Year:
2010

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Abstract

Let F be a separable Banach space, and let (X, Y) be a random pair taking values in F × R. Motivated by a broad range of potential applications, we investigate rates of convergence of the k-nearest neighbor estimate rn(X) of the regression function r(X) = E[Y|X = X], based on n independent copies of the pair (X, Y). Using compact embedding theory, we present explicit and general finite sample bounds on the expected squared difference E[rn(X) - r(X)]2, and particularize our results to classical function spaces such as Sobolev spaces, Besov spaces, and reproducing kernel Hilbert spaces.