Sets of approximating functions with finite Vapnik–Chervonenkis dimension for nearest-neighbors algorithms

► Reformulation of k-NN algorithm to alpha-NN ∗ algorithm (where alpha is a fraction). ► Establishing sets of functions for alpha-NN ∗, with finite capacity. ► Pointing out degrees of freedom for these sets. ► Proving theorems about dichotomies and VC-dimension for the proposed sets. According to a...

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Veröffentlicht in:	Pattern recognition letters Jg. 32; H. 14; S. 1882 - 1893
Hauptverfasser:	KLCSK, P, KORZEN, M
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Amsterdam Elsevier B.V 15.10.2011 Elsevier
Schlagworte:	Applied sciences Complexity selection Exact sciences and technology Generalization Information, signal and communications theory k-Nearest neighbors Signal and communications theory Signal representation. Spectral analysis Signal, noise Statistical learning theory Structural risk minimization Telecommunications and information theory Vapnik–Chervonenkis dimension Statistical learning theory k-Nearest neighbors Vapnik–Chervonenkis dimension Generalization Complexity selection Structural risk minimization Learning Nearest neighbour Vapnik-Chervonenkis dimension Cross validation Algorithm Signal analysis
ISSN:	0167-8655, 1872-7344
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Abstract	► Reformulation of k-NN algorithm to alpha-NN ∗ algorithm (where alpha is a fraction). ► Establishing sets of functions for alpha-NN ∗, with finite capacity. ► Pointing out degrees of freedom for these sets. ► Proving theorems about dichotomies and VC-dimension for the proposed sets. According to a certain misconception sometimes met in the literature: for the nearest-neighbors algorithms there is no fixed hypothesis class of limited Vapnik–Chervonenkis dimension. In the paper a simple reformulation (not a modification) of the nearest-neighbors algorithm is shown where instead of a natural number k, a percentage α ∈ (0, 1) of nearest neighbors is used. Owing to this reformulation one can construct sets of approximating functions, which we prove to have finite VC dimension. In a special (but practical) case this dimension is equal to ⌊2/ α⌋. It is also then possible to form a sequence of sets of functions with increasing VC dimension, and to perform complexity selection via cross-validation or similarly to the structural risk minimization framework. Results of such experiments are also presented.
AbstractList	► Reformulation of k-NN algorithm to alpha-NN ∗ algorithm (where alpha is a fraction). ► Establishing sets of functions for alpha-NN ∗, with finite capacity. ► Pointing out degrees of freedom for these sets. ► Proving theorems about dichotomies and VC-dimension for the proposed sets. According to a certain misconception sometimes met in the literature: for the nearest-neighbors algorithms there is no fixed hypothesis class of limited Vapnik–Chervonenkis dimension. In the paper a simple reformulation (not a modification) of the nearest-neighbors algorithm is shown where instead of a natural number k, a percentage α ∈ (0, 1) of nearest neighbors is used. Owing to this reformulation one can construct sets of approximating functions, which we prove to have finite VC dimension. In a special (but practical) case this dimension is equal to ⌊2/ α⌋. It is also then possible to form a sequence of sets of functions with increasing VC dimension, and to perform complexity selection via cross-validation or similarly to the structural risk minimization framework. Results of such experiments are also presented.
Author	Korzeń, M. Klęsk, P.
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Cites_doi	10.1126/science.290.5500.2323 10.1109/TIT.1970.1054466 10.1109/5.58325 10.1145/355744.355745 10.1145/238061.238070 10.1145/361002.361007 10.1162/089976699300016304
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Keywords	Statistical learning theory k-Nearest neighbors Vapnik–Chervonenkis dimension Generalization Complexity selection Structural risk minimization Learning Nearest neighbour Vapnik-Chervonenkis dimension Cross validation Algorithm Signal analysis
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Snippet	► Reformulation of k-NN algorithm to alpha-NN ∗ algorithm (where alpha is a fraction). ► Establishing sets of functions for alpha-NN ∗, with finite capacity. ►...
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SubjectTerms	Applied sciences Complexity selection Exact sciences and technology Generalization Information, signal and communications theory k-Nearest neighbors Signal and communications theory Signal representation. Spectral analysis Signal, noise Statistical learning theory Structural risk minimization Telecommunications and information theory Vapnik–Chervonenkis dimension
Title	Sets of approximating functions with finite Vapnik–Chervonenkis dimension for nearest-neighbors algorithms
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