Double-edge factor graphs: Definition, properties, and examples

Some of the most interesting quantities associated with a factor graph are its marginals and its partition sum. For factor graphs 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 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 kernel functions. 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.