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Many iterative and non-iterative methods have been developed for inverseproblems associated with Ising models. Aiming to derive an accuratenon-iterative method for the inverse problems, we employ the tree-reweightedapproximation. Using the tree-reweighted approximation, we can optimize therigorous lower bound of the objective function. By solving the moment-matchingand self-consistency conditions analytically, we can derive the interactionmatrix as a function of the given data statistics. With this solution, we canobtain the optimal interaction matrix without iterative computation. Toevaluate the accuracy of the proposed inverse formula, we compared our resultsto those obtained by existing inverse formulae derived with otherapproximations. In an experiment to reconstruct the interaction matrix, wefound that the proposed formula returns the best estimates instrongly-attractive regions for various graph structures. We also performed anexperiment using real-world biological data. When applied to finding theconnectivity of neurons from spike train data, the proposed formula gave theclosest result to that obtained by a gradient ascent algorithm, which typicallyrequires thousands of iterations.

An Analytic Solution to the Inverse Ising Problem in the Tree-reweighted Approximation