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We propose an efficient algorithm to solve inverse problems in the presenceof binary clustered datasets. We consider the paradigmatic Hopfield model in ateacher student scenario, where this situation is found in the retrieval phase.This problem has been widely analyzed through various methods such asmean-field approaches or the pseudo-likelihood optimization. Our approach isbased on the estimation of the posterior using the Thouless-Anderson-Palmer(TAP) equations in a parallel updating scheme. At the difference with othermethods, it allows to retrieve the exact patterns of the teacher and theparallel update makes it possible to apply it for large system sizes. We alsoobserve that the Approximate Message Passing (AMP) equations do not reproducethe expected behavior in the direct problem, questioning the standard practiceused to obtain time indexes coming from Belief Propagation (BP). We tackle thesame problem using a Restricted Boltzmann Machine (RBM) and discuss theanalogies between the two algorithms.

Inverse problems for structured datasets using parallel TAP equations and RBM