Hansen, Toke Jansen3; Abrahamsen, Trine Julie1; Hansen, Lars Kai4
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 Department of Informatics and Mathematical Modeling, Technical University of Denmark4 Copenhagen Center for Health Technology, Center, Technical University of Denmark
Kernel Principal Component Analysis (PCA) has proven a powerful tool for nonlinear feature extraction, and is often applied as a pre-processing step for classification algorithms. In denoising applications Kernel PCA provides the basis for dimensionality reduction, prior to the so-called pre-image problem where denoised feature space points are mapped back into input space. This problem is inherently ill-posed due to the non-bijective feature space mapping. We present a semi-supervised denoising scheme based on kernel PCA and the pre-image problem, where class labels on a subset of the data points are used to improve the denoising. Moreover, by warping the Reproducing Kernel Hilbert Space (RKHS) we also account for the intrinsic manifold structure yielding a Kernel PCA basis that also benefit from unlabeled data points. Our two main contributions are; (1) a generalization of Kernel PCA by incorporating a loss term, leading to an iterative algorithm for finding orthonormal components biased by the class labels, and (2) a fixed-point iteration for solving the pre-image problem based on a manifold warped RKHS. We prove viability of the proposed methods on both synthetic data and images from The Amsterdam Library of Object Images (Geusebroek et al., 2005) .
Pattern Recognition Letters, 2014, Vol 49, p. 114-120
Semi-supervised denoising; Kernel PCA; Pre-image problem