  Many iterative and non-iterative methods have been developed for inverse
problems associated with Ising models. Aiming to derive an accurate
non-iterative method for the inverse problems, we employ the tree-reweighted
approximation. Using the tree-reweighted approximation, we can optimize the
rigorous lower bound of the objective function. By solving the moment-matching
and self-consistency conditions analytically, we can derive the interaction
matrix as a function of the given data statistics. With this solution, we can
obtain the optimal interaction matrix without iterative computation. To
evaluate the accuracy of the proposed inverse formula, we compared our results
to those obtained by existing inverse formulae derived with other
approximations. In an experiment to reconstruct the interaction matrix, we
found that the proposed formula returns the best estimates in
strongly-attractive regions for various graph structures. We also performed an
experiment using real-world biological data. When applied to finding the
connectivity of neurons from spike train data, the proposed formula gave the
closest result to that obtained by a gradient ascent algorithm, which typically
requires thousands of iterations.
