The connection between regularization operators and support vector kernels

Alex J. Smola; Bernhard Schölkopf; Klaus Robert Müller

doi:10.1016/S0893-6080(98)00032-X

The connection between regularization operators and support vector kernels

Alex J. Smola, Bernhard Schölkopf, Klaus Robert Müller

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

526 Citations (Scopus)

Abstract

In this paper a correspondence is derived between regularization operators used in regularization networks and support vector kernels. We prove that the Green's Functions associated with regularization operators are suitable support vector kernels with equivalent regularization properties. Moreover, the paper provides an analysis of currently used support vector kernels in the view of regularization theory and corresponding operators associated with the classes of both polynomial kernels and translation invariant kernels. The latter are also analyzed on periodical domains. As a by-product we show that a large number of radial basis functions, namely conditionally positive definite functions, may be used as support vector kernels.

Original language	English
Pages (from-to)	637-649
Number of pages	13
Journal	Neural Networks
Volume	11
Issue number	4
DOIs	https://doi.org/10.1016/S0893-6080(98)00032-X
Publication status	Published - 1998 Jun

Bibliographical note

Funding Information:
The authors thank Chris Burges, Federico Girosi, Leo van Hemmen, Takashi Onoda, John Shawe-Taylor, Vladimir Vapnik, Grace Wahba, and Alan Yuille for helpful discussions and comments. A.J. Smola is supported by a grant from the DFG (JA 379/71), and B. Schölkopf is supported by a grant from the Studienstiftung des deutschen Volkes.

Keywords

Conditionally positive definite functions
Green's functions
Mercer kernel
Polynomial kernels
Radial basis functions
Regularization networks
Ridge regression
Support vector machines

ASJC Scopus subject areas

Cognitive Neuroscience
Artificial Intelligence

Access to Document

10.1016/S0893-6080(98)00032-X

Cite this

@article{16c4011aabc648218f066dd1a40f1ae9,

title = "The connection between regularization operators and support vector kernels",

abstract = "In this paper a correspondence is derived between regularization operators used in regularization networks and support vector kernels. We prove that the Green's Functions associated with regularization operators are suitable support vector kernels with equivalent regularization properties. Moreover, the paper provides an analysis of currently used support vector kernels in the view of regularization theory and corresponding operators associated with the classes of both polynomial kernels and translation invariant kernels. The latter are also analyzed on periodical domains. As a by-product we show that a large number of radial basis functions, namely conditionally positive definite functions, may be used as support vector kernels.",

keywords = "Conditionally positive definite functions, Green's functions, Mercer kernel, Polynomial kernels, Radial basis functions, Regularization networks, Ridge regression, Support vector machines",

author = "Smola, {Alex J.} and Bernhard Sch{\"o}lkopf and M{\"u}ller, {Klaus Robert}",

note = "Funding Information: The authors thank Chris Burges, Federico Girosi, Leo van Hemmen, Takashi Onoda, John Shawe-Taylor, Vladimir Vapnik, Grace Wahba, and Alan Yuille for helpful discussions and comments. A.J. Smola is supported by a grant from the DFG (JA 379/71), and B. Sch{\"o}lkopf is supported by a grant from the Studienstiftung des deutschen Volkes. ",

year = "1998",

month = jun,

doi = "10.1016/S0893-6080(98)00032-X",

language = "English",

volume = "11",

pages = "637--649",

journal = "Neural Networks",

issn = "0893-6080",

publisher = "Elsevier Limited",

number = "4",

}

TY - JOUR

T1 - The connection between regularization operators and support vector kernels

AU - Smola, Alex J.

AU - Schölkopf, Bernhard

AU - Müller, Klaus Robert

N1 - Funding Information: The authors thank Chris Burges, Federico Girosi, Leo van Hemmen, Takashi Onoda, John Shawe-Taylor, Vladimir Vapnik, Grace Wahba, and Alan Yuille for helpful discussions and comments. A.J. Smola is supported by a grant from the DFG (JA 379/71), and B. Schölkopf is supported by a grant from the Studienstiftung des deutschen Volkes.

PY - 1998/6

Y1 - 1998/6

N2 - In this paper a correspondence is derived between regularization operators used in regularization networks and support vector kernels. We prove that the Green's Functions associated with regularization operators are suitable support vector kernels with equivalent regularization properties. Moreover, the paper provides an analysis of currently used support vector kernels in the view of regularization theory and corresponding operators associated with the classes of both polynomial kernels and translation invariant kernels. The latter are also analyzed on periodical domains. As a by-product we show that a large number of radial basis functions, namely conditionally positive definite functions, may be used as support vector kernels.

AB - In this paper a correspondence is derived between regularization operators used in regularization networks and support vector kernels. We prove that the Green's Functions associated with regularization operators are suitable support vector kernels with equivalent regularization properties. Moreover, the paper provides an analysis of currently used support vector kernels in the view of regularization theory and corresponding operators associated with the classes of both polynomial kernels and translation invariant kernels. The latter are also analyzed on periodical domains. As a by-product we show that a large number of radial basis functions, namely conditionally positive definite functions, may be used as support vector kernels.

KW - Conditionally positive definite functions

KW - Green's functions

KW - Mercer kernel

KW - Polynomial kernels

KW - Radial basis functions

KW - Regularization networks

KW - Ridge regression

KW - Support vector machines

UR - http://www.scopus.com/inward/record.url?scp=0032098361&partnerID=8YFLogxK

U2 - 10.1016/S0893-6080(98)00032-X

DO - 10.1016/S0893-6080(98)00032-X

M3 - Article

AN - SCOPUS:0032098361

SN - 0893-6080

VL - 11

SP - 637

EP - 649

JO - Neural Networks

JF - Neural Networks

IS - 4

ER -

The connection between regularization operators and support vector kernels

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this