TY - JOUR
T1 - Approximation with polynomial kernels and SVM classifiers
AU - Zhou, Ding-Xuan
AU - Jetter, Kurt
PY - 2006/7
Y1 - 2006/7
N2 - This paper presents an error analysis for classification algorithms generated by regularization schemes with polynomial kernels. Explicit convergence rates are provided for support vector machine (SVM) soft margin classifiers. The misclassification error can be estimated by the sum of sample error and regularization error. The main difficulty for studying algorithms with polynomial kernels is the regularization error which involves deeply the degrees of the kernel polynomials. Here we overcome this difficulty by bounding the reproducing kernel Hilbert space norm of Durrmeyer operators, and estimating the rate of approximation by Durrmeyer operators in a weighted L 1 space (the weight is a probability distribution). Our study shows that the regularization parameter should decrease exponentially fast with the sample size, which is a special feature of polynomial kernels. © Springer 2006.
AB - This paper presents an error analysis for classification algorithms generated by regularization schemes with polynomial kernels. Explicit convergence rates are provided for support vector machine (SVM) soft margin classifiers. The misclassification error can be estimated by the sum of sample error and regularization error. The main difficulty for studying algorithms with polynomial kernels is the regularization error which involves deeply the degrees of the kernel polynomials. Here we overcome this difficulty by bounding the reproducing kernel Hilbert space norm of Durrmeyer operators, and estimating the rate of approximation by Durrmeyer operators in a weighted L 1 space (the weight is a probability distribution). Our study shows that the regularization parameter should decrease exponentially fast with the sample size, which is a special feature of polynomial kernels. © Springer 2006.
KW - Approximation by Durrmeyer operators
KW - Classification algorithm
KW - Misclassification error
KW - Polynomial kernel
KW - Regularization scheme
KW - Support vector machine
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UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-33745650526&origin=recordpage
U2 - 10.1007/s10444-004-7206-2
DO - 10.1007/s10444-004-7206-2
M3 - 21_Publication in refereed journal
VL - 25
SP - 323
EP - 344
JO - Advances in Computational Mathematics
JF - Advances in Computational Mathematics
SN - 1019-7168
IS - 1-3
ER -