Some methods of constructing kernels in statistical learning

Tomasz Górecki; Maciej Łuczak

Discussiones Mathematicae Probability and Statistics (2010)

  • Volume: 30, Issue: 2, page 179-201
  • ISSN: 1509-9423

Abstract

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This paper is a collection of numerous methods and results concerning a design of kernel functions. It gives a short overview of methods of building kernels in metric spaces, especially R n and S n . However we also present a new theory. Introducing kernels was motivated by searching for non-linear patterns by using linear functions in a feature space created using a non-linear feature map.

How to cite

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Tomasz Górecki, and Maciej Łuczak. "Some methods of constructing kernels in statistical learning." Discussiones Mathematicae Probability and Statistics 30.2 (2010): 179-201. <http://eudml.org/doc/277015>.

@article{TomaszGórecki2010,
abstract = {This paper is a collection of numerous methods and results concerning a design of kernel functions. It gives a short overview of methods of building kernels in metric spaces, especially $R^n$ and $S^n$. However we also present a new theory. Introducing kernels was motivated by searching for non-linear patterns by using linear functions in a feature space created using a non-linear feature map.},
author = {Tomasz Górecki, Maciej Łuczak},
journal = {Discussiones Mathematicae Probability and Statistics},
keywords = {positive definite kernel; dot product kernel; statistical kernel; SVM; kPCA},
language = {eng},
number = {2},
pages = {179-201},
title = {Some methods of constructing kernels in statistical learning},
url = {http://eudml.org/doc/277015},
volume = {30},
year = {2010},
}

TY - JOUR
AU - Tomasz Górecki
AU - Maciej Łuczak
TI - Some methods of constructing kernels in statistical learning
JO - Discussiones Mathematicae Probability and Statistics
PY - 2010
VL - 30
IS - 2
SP - 179
EP - 201
AB - This paper is a collection of numerous methods and results concerning a design of kernel functions. It gives a short overview of methods of building kernels in metric spaces, especially $R^n$ and $S^n$. However we also present a new theory. Introducing kernels was motivated by searching for non-linear patterns by using linear functions in a feature space created using a non-linear feature map.
LA - eng
KW - positive definite kernel; dot product kernel; statistical kernel; SVM; kPCA
UR - http://eudml.org/doc/277015
ER -

References

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  2. [2] B.E. Boser, I.M. Guyon and V.N. Guyon, A training algorithm for optimal margin classifiers, in D. Haussler, eds. 5th Annual ACM Workshop on COLT. ACM Press, Pittsburgh (1992), 144-152. 
  3. [3] C.J.C. Burges, Geometry and invariance in kernel based methods in: Schölkopf, B. Burges, C.J.C. Smola, A.J. eds. Advances in kernel methods - support vector learning. MIT Press, Cambridge (1999), 89-116. 
  4. [4] C. Cortes and V. Vapnik, Support-Vector Networks, Machine Learning 20 (1995), 273-297. 
  5. [5] R. Herbrich, Learning Kernel Classifiers, MIT Press, London 2002. Zbl1063.62092
  6. [6] T. Hofmann, B. Schölkopf and A.J. Smola, Kernels methods in machine learning, Annals of Statistics 36 (2008), 1171-1220. Zbl1151.30007
  7. [7] Z. Ovari, Kernels, eigenvalues and support vector machines, Honours thesis, Australian National University, Canberra 2000. 
  8. [8] B. Schölkopf and A.J. Smola, Learning with Kernels, MIT Press, London 2002. 
  9. [9] B. Schölkopf, A.J. Smola and K.R. Müller, Nonlinear component analysis as a kernel eigenvalue problem, Neural Computation 10 (1998), 1299-1319. 
  10. [10] I.J. Schoenberg, Positive definite functions on spheres, Duke Mathematical Journal 9 (1942), 96-108. Zbl0063.06808
  11. [11] A. Tarantola, Inverse problem theory and methods for model paramenter estimation, SIAM, Philadelphia 2005. Zbl1074.65013
  12. [12] M. Zu, Kernels and ensembles: perspective on statistical learning, American Statistician 62 (2008), 97-109. 

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