RKHS to find the Representer Theorem for regularization operators whose null space is not finite dimensional
Harshit Gupta, EPFL STI LIB

Meeting • 09 February 2017

Abstract
We will discuss the use of the theory of Reproducing Kernel Hilbert Spaces to find the Representer Theorem for regularization operators (a few) whose null space is not finite dimensional. Representer Theorems (RT) deal with the parametric representation of the solution of a regularized linear inverse problem. Any RT needs a proper specification of regularization and its operator, and the search space (with both regularized and unregularized components). Extending the RT to higher dimensions pose the problem of unavailabilty of operators with finite dimensional null-space (a necessary assumption). Using the theory of Reproducing Kernel in Hilbert Space can be advantageous in this sense. Using this theory helps in resolving the problem of finding the search space (native space) for a given operator even when the null space is not finite dimensional. This is done by including only a finite dimensional part of the null space in the search space. The Reproducing kernel of this space is essential in defining the search space itself and finding the parametric form of the solution.