Nystr\"om landmark sampling and regularized Christoffel functions

Selecting diverse and important items from a large set is a problem of interest in machine learning. As a specific example, in order to deal with large training sets, kernel methods often rely on low rank matrix approximations based on the selection or sampling of Nystr\"om centers. In this context, we propose a deterministic and a randomized adaptive algorithm for selecting landmark points within a training dataset, which are related to the minima of a sequence of Christoffel functions in Reproducing Kernel Hilbert Spaces. Beyond the known connection between Christoffel functions and leverage scores, a connection of our method with determinantal point processes (DPP) is also explained. Namely, our construction promotes diversity among important landmark points in a way similar to DPPs.