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We consider large-scale linear inverse problems in Bayesian settings. Wefollow a recent line of work that applies the approximate message passing (AMP)framework to multi-processor (MP) computational systems, where each processornode stores and processes a subset of rows of the measurement matrix along withcorresponding measurements. In each MP-AMP iteration, nodes of the MP systemand its fusion center exchange lossily compressed messages pertaining to theirestimates of the input. In this setup, we derive the optimal per-iterationcoding rates using dynamic programming. We analyze the excess mean squarederror (EMSE) beyond the minimum mean squared error (MMSE), and prove that, inthe limit of low EMSE, the optimal coding rates increase approximately linearlyper iteration. Additionally, we obtain that the combined cost of computationand communication scales with the desired estimation quality according to$O(\log^2(1/\text{EMSE}))$. Finally, we study trade-offs between the physicalcosts of the estimation process including computation time, communicationloads, and the estimation quality as a multi-objective optimization problem,and characterize the properties of the Pareto optimal surfaces.

Optimal Trade-offs in Multi-Processor Approximate Message Passing