  We introduce and study a mathematical framework for a broad class of
regularization functionals for ill-posed inverse problems: Regularization
Graphs. Regularization graphs allow to construct functionals using as building
blocks linear operators and convex functionals, assembled by means of operators
that can be seen as generalizations of classical infimal convolution operators.
This class of functionals exhaustively covers existing regularization
approaches and it is flexible enough to craft new ones in a simple and
constructive way. We provide well-posedness and convergence results with the
proposed class of functionals in a general setting. Further, we consider a
bilevel optimization approach to learn optimal weights for such regularization
graphs from training data. We demonstrate that this approach is capable of
optimizing the structure and the complexity of a regularization graph,
allowing, for example, to automatically select a combination of regularizers
that is optimal for given training data.
