Orbit-Product Representation and Correction of Gaussian Belief Propagation

We present a new view of Gaussian belief propagation (GaBP) based on a representa- tion of the determinant as a product over or- bits of a graph. We show that the GaBP determinant estimate captures totally back- tracking orbits of the graph and consider how to correct this estimate. We show that the missing orbits may be grouped into equiva- lence classes corresponding to backtrackless orbits and the contribution of each equiv- alence class is easily determined from the GaBP solution. Furthermore, we demon- strate that this multiplicative correction fac- tor can be interpreted as the determinant of a backtrackless adjacency matrix of the graph with edge weights based on GaBP. Finally, an efficient method is proposed to compute a truncated correction factor including all backtrackless orbits up to a specified length.