1 Department of Applied Mathematics and Computer Science, Technical University of Denmark2 Cognitive Systems, Department of Applied Mathematics and Computer Science, Technical University of Denmark3 Copenhagen Center for Health Technology, Center, Technical University of Denmark
Kernel methods refer to a family of widely used nonlinear algorithms for machine learning tasks like classification, regression, and feature extraction. By exploiting the so-called kernel trick straightforward extensions of classical linear algorithms are enabled as long as the data only appear as innerproducts in the model formulation. This dissertation presents research on improving the performance of standard kernel methods like kernel Principal Component Analysis and the Support Vector Machine. Moreover, the goal of the thesis has been two-fold. The first part focuses on the use of kernel Principal Component Analysis for nonlinear denoising. In this context stable solution of the inverse and inherently ill-posed pre-image problem constitutes the main challenge. It is proposed to stabilize the estimation by augmenting the cost function with either an `1-or `2-norm penalty, and solution schemes are derived for both approaches. The methods are experimentally validated on several biomedical data sets. Furthermore, frameworks for exploiting label information for improved denoising in the semisupervised case are proposed. The second part of the thesis examines the effect of variance inflation in kernel methods. Variance inflation occurs in high-dimensional problems when the training data are insufficient to describe the entire signal manifold. Thereby leading to a potential mismatch between the subspaces spanned by the training and test data, respectively. It is shown how this effect extends from linear models to kernel learning, and means for restoring the generalizability in both kernel Principal Component Analysis and the Support Vector Machine are proposed. Viability is proved on a wide range of benchmark machine learning data sets.