The Max-Product Algorithm Viewed as Linear Data-Fusion: A Distributed Detection Scenario

TL;DR

This paper analyzes the max-product algorithm in distributed hypothesis testing, revealing it behaves like a linear data-fusion scheme similar to the sum-product algorithm, and demonstrates optimal performance through linear combinations of local likelihoods.

Contribution

It shows that the max-product algorithm's performance can be achieved by a linear data-fusion scheme, linking it closely to the sum-product algorithm's behavior in distributed detection.

Findings

Max-product decision variables are linear combinations of local log-likelihood ratios.

Optimal max-product performance is achieved by a linear data-fusion scheme.

Simulation results confirm the theoretical analysis.

Abstract

In this paper, we disclose the statistical behavior of the max-product algorithm configured to solve a maximum a posteriori (MAP) estimation problem in a network of distributed agents. Specifically, we first build a distributed hypothesis test conducted by a max-product iteration over a binary-valued pairwise Markov random field and show that the decision variables obtained are linear combinations of the local log-likelihood ratios observed in the network. Then, we use these linear combinations to formulate the system performance in terms of the false-alarm and detection probabilities. Our findings indicate that, in the hypothesis test concerned, the optimal performance of the max-product algorithm is obtained by an optimal linear data-fusion scheme and the behavior of the max-product algorithm is very similar to the behavior of the sum-product algorithm. Consequently, we demonstrate that the optimal performance of the max-product iteration is closely achieved via a linear version of the sum-product algorithm which is optimized based on statistics received at each node from its one-hop neighbors. Finally, we verify our observations via computer simulations.