Part IV - Kernel ridge regressors and variants

Summary

This part contains three chapters, covering learning models closely related to the kernel ridge regressor (KRR).

1. (i) Chapter 7 discusses applications of the kernel ridge regressor (KRR) to regularization and approximation problems.

2. (ii) Chapter 8 covers a broad spectrum of conventional linear classification techniques, all related to ridge regression.

3. (iii) Chapter 9 extends the linear classification to kernel methods for nonlinear classifiers that are closely related to the kernel ridge regressor (KRR).

Chapter 7 discusses the role of ridge regression analysis in regularization problems. The optimization formulation for the kernel ridge regressor (KRR), in the intrinsic space, obviously satisfies the LSP condition in Theorem 1.1, implying the existence of a kernelized KRR learning model, see Algorithm 7.1.

If a Gaussian RBF kernel is adopted, the KRR learning model is intimately related to several prominent regularization models:

1. (i) the RBF approximation networks devised by Poggio and Girosi [207];

2. (ii) Nadaraya–Watson regression estimators [191, 293]; and

3. (iii) RBF-based back-propagation neural networks [198, 226, 294].

The chapter also addresses the role of the multi-kernel formulation in regression/classification problems. It shows that, if the LSP condition is satisfied, then some multi-kernel problems effectively warant a mono-kernel solution.

Chapter 8 covers a family of linear classifiers. The basic linear regression analysis leads to the classical least-squares-error (LSE) classifiers, see Algorithm 8.1.The derivation of Fisher's conventional linear discriminant analysis is based on finding the best projection direction which can maximize the so-called signal-to-noise ratio; see Algorithm 8.2.