Orbit-Product Representation and Correction of Gaussian Belief Propagation

We present a new view of Gaussian belief propagation (GaBP) based on a representation of the determinant as a product over orbits of a graph. We show that the GaBP determinant estimate captures totally backtracking orbits of the graph and consider how to correct this estimate. We show that the missing orbits may be grouped into equivalence classes corresponding to backtrackless orbits and the contribution of each equivalence class is easily determined from the GaBP solution. Furthermore, we demonstrate that this multiplicative correction factor 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.

💡 Research Summary

The paper introduces a novel perspective on Gaussian Belief Propagation (GaBP) by expressing the determinant of a Gaussian graphical model as a product over graph orbits. The authors first show that the determinant estimate produced by standard GaBP corresponds exactly to the contribution of totally backtracking orbits—closed walks that immediately reverse direction at each step. Because these backtracking walks ignore the genuine cyclic structure of the graph, the GaBP estimate can be substantially biased, especially in loopy or dense networks.

To remedy this, the authors categorize the missing orbits into backtrackless (or non‑backtracking) orbits and further group them into equivalence classes. Each class contains all walks that share the same underlying simple cycle but differ in starting point or orientation. Remarkably, the total contribution of an equivalence class can be computed directly from the GaBP solution: after GaBP converges, each edge acquires an “effective weight” (\hat w_e). The weight of a class (C) is simply the product of the effective weights of the edges that compose the class, (\hat w_C = \prod_{e\in C}\hat w_e). Consequently, the exact determinant can be written as

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