On the Effect of Centering Kernels in Kernel PCA

• Activity Number: 366
• Type: Contributed
• Date/Time: Tuesday, August 6, 2013 : 10:30 AM to 12:20 PM
• Sponsor: Section on Statistical Learning and Data Mining
• Author(s): Zhiyu Liang*+ and Yoonkyung Lee
• Companies: The Ohio State University and The Ohio State University
• Keywords: centered kernel ; Gaussian kernel ; kernel methods ; kernel PCA ; polynomial kernel

Abstract

Kernel principal component analysis (Kernel PCA) is widely used in statistical learning and data mining applications. The eigen-analysis of the kernel operator provides insights into how the kernel PCA works as a nonlinear generalization of PCA in relation to data distributions. Analogous to a covariance matrix for the standard PCA, a kernel matrix is often centered first and the centered version is used as an input for kernel PCA. The eigen-analysis of the centered kernel operator generally yields different results from the uncentered counterpart. In this paper, we study the effect of centering kernels, making a contrast with the effect of centering data. We focus on two commonly used kernels, namely, Gaussian kernel and polynomial kernel, and examine the difference between the centered and uncentered kernel operators. The results shed light on difference in the geometry of projections for the two versions, which reveals key features of variation in the data such as clusters.