A perturbation based out-of-sample extension framework

Out-of-sample extension is an important task in various kernel based non-linear dimensionality reduction algorithms. In this paper, we derive a perturbation based extension framework by extending results from classical perturbation theory. We prove that our extension framework generalizes the well-known Nystr{\"o}m method as well as some of its variants. We provide an error analysis for our extension framework, and suggest new forms of extension under this framework that take advantage of the structure of the kernel matrix. We support our theoretical results numerically and demonstrate the advantages of our extension framework both on synthetic and real data.