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Approximate message passing (AMP) is a class of low-complexity, scalablealgorithms for solving high-dimensional linear regression tasks where onewishes to recover an unknown signal from noisy, linear measurements. AMP is aniterative algorithm that performs estimation by updating an estimate of theunknown signal at each iteration and the performance of AMP (quantified, forexample, by the mean squared error of its estimates) depends on the choice of a"denoiser" function that is used to produce these signal estimates at eachiteration.  An attractive feature of AMP is that its performance can be tracked by ascalar recursion referred to as state evolution. Previous theoretical analysisof the accuracy of the state evolution predictions has been limited to the useof only separable denoisers or block-separable denoisers, a class of denoisersthat underperform when sophisticated dependencies exist between signal entries.Since signals with entrywise dependencies are common in image/video-processingapplications, in this work we study the high-dimensional linear regression taskwhen the dependence structure of the input signal is modeled by a Markov randomfield prior distribution. We provide a rigorous analysis of the performance ofAMP, demonstrating the accuracy of the state evolution predictions, when aclass of non-separable sliding-window denoisers is applied. Moreover, weprovide numerical examples where AMP with sliding-window denoisers cansuccessfully capture local dependencies in images.

Analysis of Approximate Message Passing with Non-Separable Denoisers and Markov Random Field Priors