Decentralized Generalized Approximate Message-Passing for Tree-Structured Networks

TL;DR

This work tackles decentralized signal reconstruction for distributed generalized linear measurements on tree networks by introducing D-GAMP, which couples local GAMP iterations with consensus propagation. The authors develop a rigorous state evolution framework that characterizes the asymptotic behavior of the decentralized algorithm and prove that, under homogeneous measurements, its fixed points match those of the centralized GAMP. They further show convergence of SE when Bayes-optimal denoisers are used via a long-memory proof, and validate the theory with numerical results for linear and clipping measurements on chain and tree topologies. The approach enables scalable, distributed compressed sensing without a central fusion center while preserving Bayes-optimal performance at the fixed point, provided unique convergence. Limitations include the need for zero-mean i.i.d. sensing matrices and tree-structured networks, suggesting future work to extend to general ad hoc graphs and alternative consensus strategies.

Abstract

Decentralized generalized approximate message-passing (GAMP) is proposed for compressed sensing from distributed generalized linear measurements in a tree-structured network. Consensus propagation is used to realize average consensus required in GAMP via local communications between adjacent nodes. Decentralized GAMP is applicable to all tree-structured networks that do not necessarily have central nodes connected to all other nodes. State evolution is used to analyze the asymptotic dynamics of decentralized GAMP for zero-mean independent and identically distributed Gaussian sensing matrices. The state evolution recursion for decentralized GAMP is proved to have the same fixed points as that for centralized GAMP when homogeneous measurements with an identical dimension in all nodes are considered. Furthermore, existing long-memory proof strategy is used to prove that the state evolution recursion for decentralized GAMP with the Bayes-optimal denoisers converges to a fixed point. These results imply that the state evolution recursion for decentralized GAMP with the Bayes-optimal denoisers converges to the Bayes-optimal fixed point for the homogeneous measurements when the fixed point is unique. Numerical results for decentralized GAMP are presented in the cases of linear measurements and clipping. As examples of tree-structured networks, a one-dimensional chain and a tree with no central nodes are considered.

Figures (5)

• Figure 1: Tree-structured networks with no central nodes.
• Figure 2: Largest MSE versus the total number of inner iterations for consensus propagation in the linear measurements. One-dimensional chain network with $L=4$ nodes, measurement dimension $M[l]=480$, signal dimension $N=6400$, signal density $\rho=0.1$, SNR $1/\sigma^{2}=30$ dB, and damping factor $\chi=1$. The solid curves show state evolution results while numerical simulations are plotted with markers.
• Figure 3: Largest MSE versus the total number of inner iterations for consensus propagation in the clipping case. One-dimensional chain network with $L=4$ nodes, measurement dimension $M[l]=800$, signal dimension $N=4000$, signal density $\rho=0.1$, threshold $A=2$, SNR $1/\sigma^{2}=30$ dB, and damping factor $\chi=1$. The solid curves show state evolution results while numerical simulations are plotted with markers. As a heterogeneous case, $T[l]=2$ for odd $l$, $T[l]=1$ for even $l$, and $J=1$ were considered.
• Figure 4: Largest MSE versus the number of iterations in the inner module for the clipping case. Tree network with $L=8$ nodes, compression ratio $M[l]/N=0.05$, signal density $\rho=0.1$, threshold $A=2$, SNR $1/\sigma^{2}=30$ dB, $T[l]=1$, and $J=1$. For $N=500, 1000, 2000$, D-GAMP used damping factors $\chi=0.9, 1, 1$, respectively, while centralized GAMP used $\chi=0.9, 0.95, 0.95$.
• Figure 5: Largest MSE versus the number of iterations in the inner module for the clipping case. Tree network with $L=8$ nodes, signal dimension $N=600$, signal density $\rho=0.1$, threshold $A=2$, SNR $1/\sigma^{2}=30$ dB, $T[l]=1$, and $J=1$. $M[l]=90$ and $\chi=0.95$ were considered in a homogeneous case while $M[l]=150$ for odd $l$, $M[l]=30$ for even $l$, and $\chi=0.95$ were considered in a heterogeneous case. Centralized GAMP used damping factor $\chi=1$.

Theorems & Definitions (26)

• Definition 1
• Definition 2: Separability
• Definition 3: Pseudo-Lipschitz
• Definition 4: Proper
• Definition 5
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma 1: Takeuchi222