An Analysis Framework for Landmark Selection and Nyström Kernel Approximation: Bounds, Algorithms, and Applications

In recent years, the spectral analysis of appropriately defined kernels has emerged as a principled way to extract the nonlinear structure inherent in vision tasks as diverse as segmentation and recognition. For computational reasons, a landmark selection process is often employed to choose a partial kernel, followed by a Nyström extension that provides its minimally positive definite completion. The literature remains open on the question of optimal landmark selection, however, and here we develop an analysis framework for this problem that subsumes previous approaches. We first show how it leads to quantitative performance bounds for both existing and new algorithms. We then discuss a range of methods for optimizing the landmark selection process through a stochastic procedure, and finally demonstrate their effectiveness in two key applications: the low-level vision task of image segmentation, and the high-level task of nonlinear dimensionality reduction from video data.