Convergence Rate of a Message-passing Algorithm for Solving Linear Systems

TL;DR

This paper analyzes the convergence rate of a generalized message-passing algorithm for large-scale linear systems, providing explicit bounds based on diagonal dominance and offering new insights into Gaussian Belief Propagation.

Contribution

It introduces a generalized convergence analysis for message-passing algorithms, extending Gaussian BP, with explicit bounds derived from linear algebraic properties.

Findings

Convergence rate bounds are explicitly derived using diagonal dominance.

The analysis applies to generalized message-passing algorithms beyond Gaussian BP.

Provides new theoretical insights into the behavior of Gaussian BP.

Abstract

This paper studies the convergence rate of a message-passing distributed algorithm for solving a large-scale linear system. This problem is generalised from the celebrated Gaussian Belief Propagation (BP) problem for statistical learning and distributed signal processing, and this message-passing algorithm is generalised from the well-celebrated Gaussian BP algorithm. Under the assumption of generalised diagonal dominance, we reveal, through painstaking derivations, several bounds on the convergence rate of the message-passing algorithm. In particular, we show clearly how the convergence rate of the algorithm can be explicitly bounded using the diagonal dominance properties of the system. When specialised to the Gaussian BP problem, our work also offers new theoretical insight into the behaviour of the BP algorithm because we use a purely linear algebraic approach for convergence analysis.