Convergence analysis of belief propagation for pairwise linear Gaussian models
This work addresses a theoretical gap for distributed inference in networks like smart grids, but it is incremental as it focuses on a specific model.
The paper tackles the open problem of convergence conditions for Gaussian belief propagation in pairwise linear Gaussian models, showing analytically that the information matrix converges geometrically and providing necessary and sufficient conditions for the belief mean to reach the optimal estimate.
Gaussian belief propagation (BP) has been widely used for distributed inference in large-scale networks such as the smart grid, sensor networks, and social networks, where local measurements/observations are scattered over a wide geographical area. One particular case is when two neighboring agents share a common observation. For example, to estimate voltage in the direct current (DC) power flow model, the current measurement over a power line is proportional to the voltage difference between two neighboring buses. When applying the Gaussian BP algorithm to this type of problem, the convergence condition remains an open issue. In this paper, we analyze the convergence properties of Gaussian BP for this pairwise linear Gaussian model. We show analytically that the updating information matrix converges at a geometric rate to a unique positive definite matrix with arbitrary positive semidefinite initial value and further provide the necessary and sufficient convergence condition for the belief mean vector to the optimal estimate.