By Johan A. K. Suykens
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Extra resources for Advances in learning theory: methods, models, and applications
Note that since K" is a Mercer kernel JFf[x] is positive semidefinite. t. /(/(x)-2/) 2 Jx / € Hx 32 F. dicker, S. t. m i=i - /eft*. For x € X, let /Q : X -+ E be given by #*(*) = #(z, t}. Theorem 1 For all 7 > 0, i/ie minimizers /7 and /7)Z 0/^(7) and $z(7) respectively exist and are unique. In addition and /7>z is given by where a= (ai, . . , am) is t/ie unique solution of the well-posed linear system in lR (Tmld + K[x])a - y. Finally, for f = Y%L\ a-iKx* we have \\f\\2K = aTK[x\a. PROOF. See Propositions 7 and 8 and Theorem 2 in Chapter III of [CS] and its references, and  and its references.
Am) is t/ie unique solution of the well-posed linear system in lR (Tmld + K[x])a - y. Finally, for f = Y%L\ a-iKx* we have \\f\\2K = aTK[x\a. PROOF. See Propositions 7 and 8 and Theorem 2 in Chapter III of [CS] and its references, and  and its references. 3 Estimating the Confidence Define, for / G £^(X), its error £(/)= Jz f(f(x)-y} and, given a sample z € Zm, its empirical error Note that from the equality £(/7)Z) = ^(/y,z) — ^(/y) + £(/>) we deduce We will call the first term in the right-hand side the sample error (this use of the expression slightly differs from the one in [CS]) and the second, the approximation error.
According to this theory, to guarantee a high rate of generalization of the learning machine one has to construct a structure S-\ C So C ... C S 12 This name stresses that for constructing this type of machine, the idea of expanding the solution on support vectors is crucial. In the SVM the complexity of construction depends on the number of support vectors rather than on the dimensionality of the feature space. 24 V. 20). 16) can be rewritten in the simple form where the first term is an estimate of the risk and the second term is the confidence interval for this estimate.
Advances in learning theory: methods, models, and applications by Johan A. K. Suykens