Reduced Set KPCA for Improving the Training and Execution Speed of Kernel Machines

This paper presents a practical, and theoretically wellfounded, approach to improve the speed of kernel manifold learning algorithms relying on spectral decomposition. Utilizing recent insights in kernel smoothing and learning with integral operators, we propose Reduced Set KPCA (RSKPCA), which also suggests an easy-toimplement method to remove or replace samples with minimal effect on the empirical operator. A simple data point selection procedure is given to generate a substitute density for the data, with accuracy that is governed by a user-tunable parameter `. The effect of the approximation on the quality of the KPCA solution, in terms of spectral and operator errors, can be shown directly in terms of the density estimate error and as a function of the parameter `. We show in experiments that RSKPCA can improve both training and evaluation time of KPCA by up to an order of magnitude, and compares favorably to the widely-used Nystrom and density-weighted Nystrom methods.