  Message passing on factor graphs is a powerful framework for probabilistic
inference, which finds important applications in various scientific domains.
The most wide-spread message passing scheme is the sum-product algorithm (SPA)
which gives exact results on trees but often fails on graphs with many small
cycles. We search for an alternative message passing algorithm that works
particularly well on such cyclic graphs. Therefore, we challenge the extrinsic
principle of the SPA, which loses its objective on graphs with cycles. We
further replace the local SPA message update rule at the factor nodes of the
underlying graph with a generic mapping, which is optimized in a data-driven
fashion. These modifications lead to a considerable improvement in performance
while preserving the simplicity of the SPA. We evaluate our method for two
classes of cyclic graphs: the 2x2 fully connected Ising grid and factor graphs
for symbol detection on linear communication channels with inter-symbol
interference. To enable the method for large graphs as they occur in practical
applications, we develop a novel loss function that is inspired by the Bethe
approximation from statistical physics and allows for training in an
unsupervised fashion.
