A graph-dynamics interpretation of the sum-product algorithm

Summary form only given: A well-known result by Yedidia, Freeman, and Weiss (IEEE Trans. Inf. Theory, vol. 51, no. 7, pp. 2282-2312, July 2005) says that fixed points of the sum-product algorithm correspond to stationary points of the (variational) Bethe free energy. This result can be given a reinterpretation in terms of graph covers. Namely, when running the sum-product algorithm on a Tanner graph then a fixed point corresponds to a certain pseudo-codeword of that Tanner graph: it is, after taking a biasing channel-output-dependent term properly into account, the pseudo-codeword that has locally the most (or the least) pre-images in all M-covers when M goes to infinity. This observation can be suitably extended to the transient part of the sum-product algorithm by expressing the sum-product algorithm in terms of a graph-dynamical system involving graph covers, twisted graph covers, and valid configurations therein. This yields new insights into the sum-product algorithm and its behavior.