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| 1 The framework of learning 1.1 Introduction 1.2 A formal setting 1.3 Hypothesis spaces and target functions 1.4 Sample, approximation, and generalization errors 1.5 The bias-variance problem 1.6 The remainder of this book 1.7 References and additional remarks 2 Basic hypothesis spaces 2.1 First examples of hypothesis space 2.2 Reminders I 2.3 Hypothesis spaces associated with Sobolev spaces 2.4 Reproducing Kernel Hilbert Spaces 2.5 Some Mercer kernels 2.6 Hypothesis spaces associated with an RKHS 2.7 Reminders II 2.8 On the computation of empirical target functions 2.9 References and additional remarks 3 Estimating the sample error 3.1 Exponential inequalities in probability 3.2 Uniform estimates on the defect 3.3 Estimating the sample error 3.4 Convex hypothesis spaces 3.5 References and additional remarks 4 Polynomial decay of the approximation error 4.1 Reminders IlI 4.2 Operators defined by a kernel 4.3 Mercer's theorem 4.4 RKHSs revisited 4.5 Characterizing the approximation error in RKHSs 4.6 An example 4.7 References and additional remarks 5 Estimating covering numbers 5.1 Reminders IV 5.2 Covering numbers for Sobolev smooth kernels 5.3 Covering numbers for analytic kernels 5.4 Lower bounds tbr covering numbers 5.5 On the smoothness of box spline kernels 5.6 References and additional remarks 6 Logarithmic decay of the approximation error 6.1 Polynomial decay of the approximation error for kernels 6.2 Measuring the regularity of the kernel 6.3 Estimating the approximation error in RKHSs 6.4 Proof of Theorem 6.1 6.5 References and additional remarks 7 On the bias-variance problem 7.1 A useful lemma 7.2 Proof of Theorem 7.1 7.3 A concrete example of bias-variance 7.4 References and additional remarks 8 Least squares regularization 8.1 Bounds for the regularized error 8.2 On the existence of target functions 8.3 A first estimate for the excess generalization error 8.4 Proof of Theorem 8.1 8.5 Reminders V 8.6 Compactness and regularization 8.7 References and additional remarks 9 Support vector machines for classification 9.1 Binary classifiers …… 10 General regularized classifiers References Index |
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