Learning sets with separating kernels

We consider the problem of learning a set from random samples. We show how relevant geometric and topological properties of a set can be studied analytically using concepts from the theory of reproducing kernel Hilbert spaces. A new kind of reproducing kernel, that we call separating kernel, plays a crucial role in our study and is analyzed in detail. We prove a new analytic characterization of the support of a distribution, that naturally leads to a family of regularized learning algorithms which are provably universally consistent and stable with respect to random sampling. Numerical experiments show that the proposed approach is competitive, and often better, than other state of the art techniques.