Accuracy-Memory Tradeoffs and Phase Transitions in Belief Propagation
This addresses a fundamental limitation in graph inference algorithms for researchers in statistical physics, bioinformatics, and network analysis, though it is an incremental theoretical advance.
The paper proves a conjecture that bounded-memory message-passing algorithms are statistically weaker than Belief Propagation for the reconstruction problem, showing their phase transition occurs strictly below the Belief Propagation threshold (Kesten-Stigum bound).
The analysis of Belief Propagation and other algorithms for the {\em reconstruction problem} plays a key role in the analysis of community detection in inference on graphs, phylogenetic reconstruction in bioinformatics, and the cavity method in statistical physics. We prove a conjecture of Evans, Kenyon, Peres, and Schulman (2000) which states that any bounded memory message passing algorithm is statistically much weaker than Belief Propagation for the reconstruction problem. More formally, any recursive algorithm with bounded memory for the reconstruction problem on the trees with the binary symmetric channel has a phase transition strictly below the Belief Propagation threshold, also known as the Kesten-Stigum bound. The proof combines in novel fashion tools from recursive reconstruction, information theory, and optimal transport, and also establishes an asymptotic normality result for BP and other message-passing algorithms near the critical threshold.