Nearest neighbor classification in infinite dimension
ESAIM: Probability and Statistics, Volume 10 (2006), pp. 340-355.

Let X be a random element in a metric space (,d), and let Y be a random variable with value 0 or 1. Y is called the class, or the label, of X. Let (X i ,Y i ) 1in be an observed i.i.d. sample having the same law as (X,Y). The problem of classification is to predict the label of a new random element X. The k-nearest neighbor classifier is the simple following rule: look at the k nearest neighbors of X in the trial sample and choose 0 or 1 for its label according to the majority vote. When (,d)=( d ,||.||), Stone (1977) proved in 1977 the universal consistency of this classifier: its probability of error converges to the Bayes error, whatever the distribution of (X,Y). We show in this paper that this result is no longer valid in general metric spaces. However, if (,d) is separable and if some regularity condition is assumed, then the k-nearest neighbor classifier is weakly consistent.

DOI: 10.1051/ps:2006014
Classification: 62H30
Keywords: classification, consistency, non parametric statistics
     author = {C\'erou, Fr\'ed\'eric and Guyader, Arnaud},
     title = {Nearest neighbor classification in infinite dimension},
     journal = {ESAIM: Probability and Statistics},
     pages = {340--355},
     publisher = {EDP-Sciences},
     volume = {10},
     year = {2006},
     doi = {10.1051/ps:2006014},
     mrnumber = {2247925},
     language = {en},
     url = {}
AU  - Cérou, Frédéric
AU  - Guyader, Arnaud
TI  - Nearest neighbor classification in infinite dimension
JO  - ESAIM: Probability and Statistics
PY  - 2006
SP  - 340
EP  - 355
VL  - 10
PB  - EDP-Sciences
UR  -
DO  - 10.1051/ps:2006014
LA  - en
ID  - PS_2006__10__340_0
ER  - 
%0 Journal Article
%A Cérou, Frédéric
%A Guyader, Arnaud
%T Nearest neighbor classification in infinite dimension
%J ESAIM: Probability and Statistics
%D 2006
%P 340-355
%V 10
%I EDP-Sciences
%R 10.1051/ps:2006014
%G en
%F PS_2006__10__340_0
Cérou, Frédéric; Guyader, Arnaud. Nearest neighbor classification in infinite dimension. ESAIM: Probability and Statistics, Volume 10 (2006), pp. 340-355. doi : 10.1051/ps:2006014.

[1] C. Abraham, G. Biau and B. Cadre, On the kernel rule for function classification. submitted (2003). | Zbl

[2] G. Biau, F. Bunea and M.H. Wegkamp, On the kernel rule for function classification. IEEE Trans. Inform. Theory, to appear (2005). | MR

[3] T.M. Cover and P.E. Hart, Nearest neighbor pattern classification. IEEE Trans. Inform. Theory IT-13 (1967) 21-27. | Zbl

[4] S. Dabo-Niang and N. Rhomari, Nonparametric regression estimation when the regressor takes its values in a metric space, submitted (2001). | Zbl

[5] L. Devroye, On the almost everywhere convergence of nonparametric regression function estimates. Ann. Statist. 9 (1981) 1310-1319. | Zbl

[6] L. Devroye, L. Györfi, A. Krzyżak and G. Lugosi, On the strong universal consistency of nearest neighbor regression function estimates. Ann. Statist. 22 (1994) 1371-1385. | Zbl

[7] L. Devroye, L. Györfi and G. Lugosi, A probabilistic theory of pattern recognition 31, Applications of Mathematics (New York). Springer-Verlag, New York (1996). | MR | Zbl

[8] L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1992). | MR | Zbl

[9] H. Federer, Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York (1969). | MR | Zbl

[10] D. Preiss, Gaussian measures and the density theorem. Comment. Math. Univ. Carolin. 22 (1981) 181-193. | Zbl

[11] D. Preiss, Dimension of metrics and differentiation of measures, in General topology and its relations to modern analysis and algebra, V (Prague, 1981), Sigma Ser. Pure Math., Heldermann, Berlin 3 (1983) 565-568. | Zbl

[12] D. Preiss and J. Tišer, Differentiation of measures on Hilbert spaces, in Measure theory, Oberwolfach 1981 (Oberwolfach, 1981), Springer, Berlin. Lect. Notes Math. 945 (1982) 194-207. | Zbl

[13] C.J. Stone, Consistent nonparametric regression. Ann. Statist. 5 (1977) 595-645. With discussion and a reply by the author. | Zbl

Cited by Sources: