Consistency and Regression with Laplacian regularization in Reproducing Kernel Hilbert Space

This note explained a way to look at reproducing kernel Hilbert space for regression problems. It consists in expressing kernel regresssion solutions with simple integral operators algebra, which we can approximate consistently from empirical data, providing the corresponding estimators of the solutions. Let's consider the classical regression problem arg min ‖f(x) y‖ In practice we are going to restrict the search for a solution f F, over a simpler function space f H. Let's associate to it the canonical RKHS, see Aronszajn (1950) H It is good to find function f from X to R, but what if Y is a real Hilbert space. Indeed, it is natural to extend the theory of RKHS to vector valued functions Schwartz (1964). Once again we can build an Hilbert space of functions from X to Y, let's first define γ Those are going to be the building element of H Definition 1 (The RKHS H).