On approximating the f-divergence between two Ising models

The f-divergence is a fundamental notion that measures the difference between two distributions. In this paper, we study the problem of approximating the f-divergence between two Ising models, which is a generalization of recent work on approximating the TV-distance. Given two Ising models $ν$ and $μ$, which are specified by their interaction matrices and external fields, the problem is to approximate the f-divergence Df(ν\,\|\,μ) within an arbitrary relative error e± ε. For χ^α-divergence with a constant integer $α$, we establish both algorithmic and hardness results. The algorithm works in a parameter regime that matches the hardness result. Our algorithm can be extended to other f-divergences such as $α$-divergence, Kullback-Leibler divergence, Rényi divergence, Jensen-Shannon divergence, and squared Hellinger distance.