VC bounds on the cardinality of nearly orthogonal function classes

Lee Ad Gottlieb; Aryeh Kontorovich; Elchanan Mossel

doi:10.1016/j.disc.2012.01.030

VC bounds on the cardinality of nearly orthogonal function classes

Lee Ad Gottlieb, Aryeh Kontorovich, Elchanan Mossel

Research output: Contribution to journal › Article › peer-review

5 Scopus citations

Abstract

We bound the number of nearly orthogonal vectors with fixed VC-dimension over {-1,1} ⁿ. Our bounds are of interest in machine learning and empirical process theory and improve previous bounds by Haussler. The bounds are based on a simple projection argument and they generalize to other product spaces. Along the way we derive tight bounds on the sum of binomial coefficients in terms of the entropy function.

Original language	English
Pages (from-to)	1766-1775
Number of pages	10
Journal	Discrete Mathematics
Volume	312
Issue number	10
DOIs	https://doi.org/10.1016/j.disc.2012.01.030
State	Published - 28 May 2012
Externally published	Yes

Keywords

Orthogonal
Packing number
VC dimension

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10.1016/j.disc.2012.01.030

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title = "VC bounds on the cardinality of nearly orthogonal function classes",

abstract = "We bound the number of nearly orthogonal vectors with fixed VC-dimension over \{-1,1\} n. Our bounds are of interest in machine learning and empirical process theory and improve previous bounds by Haussler. The bounds are based on a simple projection argument and they generalize to other product spaces. Along the way we derive tight bounds on the sum of binomial coefficients in terms of the entropy function.",

keywords = "Orthogonal, Packing number, VC dimension",

author = "Gottlieb, \{Lee Ad\} and Aryeh Kontorovich and Elchanan Mossel",

note = "Funding Information: The third author was partially supported by DMS 0548249 (CAREER) award, by ISF grant 1300/08 , by a Minerva Foundation grant and by an ERC Marie Curie Grant 2008 239317 . Funding Information: The first author{\textquoteright}s work was supported in part by The Israel Science Foundation (grant \#452/08 ), and by a Minerva grant. ",

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AU - Kontorovich, Aryeh

AU - Mossel, Elchanan

N1 - Funding Information: The third author was partially supported by DMS 0548249 (CAREER) award, by ISF grant 1300/08 , by a Minerva Foundation grant and by an ERC Marie Curie Grant 2008 239317 . Funding Information: The first author’s work was supported in part by The Israel Science Foundation (grant #452/08 ), and by a Minerva grant.

PY - 2012/5/28

Y1 - 2012/5/28

N2 - We bound the number of nearly orthogonal vectors with fixed VC-dimension over {-1,1} n. Our bounds are of interest in machine learning and empirical process theory and improve previous bounds by Haussler. The bounds are based on a simple projection argument and they generalize to other product spaces. Along the way we derive tight bounds on the sum of binomial coefficients in terms of the entropy function.

AB - We bound the number of nearly orthogonal vectors with fixed VC-dimension over {-1,1} n. Our bounds are of interest in machine learning and empirical process theory and improve previous bounds by Haussler. The bounds are based on a simple projection argument and they generalize to other product spaces. Along the way we derive tight bounds on the sum of binomial coefficients in terms of the entropy function.

KW - Orthogonal

KW - Packing number

KW - VC dimension

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VL - 312

SP - 1766

EP - 1775

JO - Discrete Mathematics

JF - Discrete Mathematics

IS - 10

ER -

VC bounds on the cardinality of nearly orthogonal function classes

Abstract

Keywords

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