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    RSB Decoupling Property of MAP Estimators

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    The large-system decoupling property of a MAP estimator is studied when it estimates the i.i.d. vector x\boldsymbol{x} from the observation y=Ax+z\boldsymbol{y}=\mathbf{A}\boldsymbol{x}+\boldsymbol{z} with A\mathbf{A} being chosen from a wide range of matrix ensembles, and the noise vector z\boldsymbol{z} being i.i.d. and Gaussian. Using the replica method, we show that the marginal joint distribution of any two corresponding input and output symbols converges to a deterministic distribution which describes the input-output distribution of a single user system followed by a MAP estimator. Under the bbRSB assumption, the single user system is a scalar channel with additive noise where the noise term is given by the sum of an independent Gaussian random variable and bb correlated interference terms. As the bbRSB assumption reduces to RS, the interference terms vanish which results in the formerly studied RS decoupling principle.Comment: 5 pages, presented in Information Theory Workshop 201
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