  A learning approach for determining which operator from a class of nonlocal
operators is optimal for the regularization of an inverse problem is
investigated. The considered class of nonlocal operators is motivated by the
use of squared fractional order Sobolev seminorms as regularization operators.
First fundamental results from the theory of regularization with local
operators are extended to the nonlocal case. Then a framework based on a
bilevel optimization strategy is developed which allows to choose nonlocal
regularization operators from a given class which i) are optimal with respect
to a suitable performance measure on a training set, and ii) enjoy particularly
favorable properties. Results from numerical experiments are also provided.
