A noniterative solution to the inverse Ising problem using a convex upper bound on the partition function

Abstract

The inverse Ising problem, or the learning of Ising models, is notoriously difficult, as evaluating the partition function has a large computational cost. To quickly solve this problem, inverse formulas using approximation methods such as the Bethe approximation have been developed. In this paper, we employ the tree-reweighted (TRW) approximation to construct a new inverse formula. An advantage of using the TRW approximation is that it provides a rigorous upper bound on the partition function, allowing us to optimize a lower bound for the learning objective function. We show that the moment-matching and self-consistency conditions can be solved analytically, and we obtain an analytic form of the approximate interaction matrix as a function of the given data statistics. Using this solution, we can compute the interaction matrix that is optimal to the approximate objective function without iterative computation. To evaluate the accuracy of the derived learning formula, we compared our formula to those obtained by other approximations. From our experiments on reconstructing interaction matrices, we found that the proposed formula gives the best estimates in models with strongly attractive interactions on various graphs.