  Some of the most interesting quantities associated with a factor graph are
its marginals and its partition sum. For factor graphs \emph{without cycles}
and moderate message update complexities, the sum-product algorithm (SPA) can
be used to efficiently compute these quantities exactly. Moreover, for various
classes of factor graphs \emph{with cycles}, the SPA has been successfully
applied to efficiently compute good approximations to these quantities. Note
that in the case of factor graphs with cycles, the local functions are usually
non-negative real-valued functions. In this paper we introduce a class of
factor graphs, called double-edge factor graphs (DE-FGs), which allow local
functions to be complex-valued and only require them, in some suitable sense,
to be positive semi-definite. We discuss various properties of the SPA when
running it on DE-FGs and we show promising numerical results for various
example DE-FGs, some of which have connections to quantum information
processing.
