A kernel view on manifold sub-sampling based on Karcher variance optimization

Abstract

In the Hilbert space reproducing the Gaussian kernel, projected data points are located on an hypersphere. Following some recent works on geodesic analysis on that particular manifold, we propose a method which purpose is to select a subset of input data by sampling the corresponding hypersphere. The selected data should represent correctly the input data, while also maximizing the diversity. We show how these two opposite objectives can be characterized in terms of Karcher variance optimization. The corresponding algorithms are defined and results are reported on toy datasets. This shows the interest of working on the kernelized festure space instead of the input space. © 2013 Springer-Verlag.