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31) SVM S AND N ONPARAMETRIC S TATISTICAL M ETHODS SVMs execute the idea of the structural risk minimization principle, where the choice of the appropriate element of the structure is deﬁned by the constant C (and a kernel parameters). 31) by controlling parameters (which depends on the training data) one guarantees asymptotic convergence of the SVM solutions to the best possible solution . 3. Support Vector Machines (SVMs) In 1980 Devroye and Wagner proved that classical nonparametric methods of density estimation are also universally consistent .
6. The Third Generation 447 I would like to add to this group two young AT&T researchers of that time: Yoav Freund and Robert Schapire, who did not directly follow the line of statistical learning theory and developed boosting technology that is close to the one discussed here [135, 136]. The third generation transformed both the area of machine learning research and the style of research. During a short period of time (less than ten years) they created a new direction in statistical learning theory: SVM and kernel methods.
23) i=1 subject to the constraints yi (yj αj K(xj , xi ) + b) ≥ 1 − ξi , ξi ≥ 0, i = 1, . . 24) yi yj αi αj K(xi , xj ) ≤ h. 25) j=1 and the constraint i,j=1 Problem 2a. 26) 436 2. 27) i=1 and the constraints 0 ≤ αi ≤ 1, i = 1, . . , . 28) Problem 3a. , . The solution of Problem 3a became the standard SVM method. In this solution only some of the coefﬁcients αi0 are different from zero. 22) are called the support vectors. 22) is the expansion on the support vectors. To construct a support vector machine one can use any (conditionally) positive deﬁnite function K(xi , xj ) creating different types of SVMs.
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