# Nearest neighbor classification in infinite dimension

Frédéric Cérou; Arnaud Guyader

ESAIM: Probability and Statistics (2006)

- Volume: 10, page 340-355
- ISSN: 1292-8100

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topCérou, Frédéric, and Guyader, Arnaud. "Nearest neighbor classification in infinite dimension ." ESAIM: Probability and Statistics 10 (2006): 340-355. <http://eudml.org/doc/249765>.

@article{Cérou2006,

abstract = {
Let X be a random element in a metric space (F,d), and let Y be a random variable with value 0 or 1. Y is
called the class, or the label, of X. Let (Xi,Yi)1 ≤ i ≤ n 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 $(\{\cal F\},d)=(\mathbb\{R\}^d,||.||)$,
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 We show in this paper that this result is no
longer valid in general metric spaces.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 (F,d) is
separable and if some regularity condition is assumed, then the
k-nearest neighbor classifier is weakly consistent.
},

author = {Cérou, Frédéric, Guyader, Arnaud},

journal = {ESAIM: Probability and Statistics},

keywords = {Classification; consistency; non parametric statistics.; classification},

language = {eng},

month = {9},

pages = {340-355},

publisher = {EDP Sciences},

title = {Nearest neighbor classification in infinite dimension },

url = {http://eudml.org/doc/249765},

volume = {10},

year = {2006},

}

TY - JOUR

AU - Cérou, Frédéric

AU - Guyader, Arnaud

TI - Nearest neighbor classification in infinite dimension

JO - ESAIM: Probability and Statistics

DA - 2006/9//

PB - EDP Sciences

VL - 10

SP - 340

EP - 355

AB -
Let X be a random element in a metric space (F,d), and let Y be a random variable with value 0 or 1. Y is
called the class, or the label, of X. Let (Xi,Yi)1 ≤ i ≤ n 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 $({\cal F},d)=(\mathbb{R}^d,||.||)$,
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 (F,d) is
separable and if some regularity condition is assumed, then the
k-nearest neighbor classifier is weakly consistent.

LA - eng

KW - Classification; consistency; non parametric statistics.; classification

UR - http://eudml.org/doc/249765

ER -

## References

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