Let @math be an undirected graph with @math edges and @math vertices. We show that @math -dimensional Ising models on @math can be learned from @math i.i.d. samples within expected total variation distance some constant factor of @math , and that this rate is optimal. We show that the same rate holds for the class of @math -dimensional multivariate normal undirected graphical models with respect to @math . We also identify the optimal rate of @math for Ising models with no external magnetic field.
Given samples from an unknown multivariate distribution @math , is it possible to distinguish whether @math is the product of its marginals versus @math being far from every product distribution? Similarly, is it possible to distinguish whether @math equals a given distribution @math versus @math and @math being far from each other? These problems of testing independence and goodness-of-fit have received enormous attention in statistics, information theory, and theoretical computer science, with sample-optimal algorithms known in several interesting regimes of parameters. Unfortunately, it has also been understood that these problems become intractable in large dimensions, necessitating exponential sample complexity. Motivated by the exponential lower bounds for general distributions as well as the ubiquity of Markov Random Fields (MRFs) in the modeling of high-dimensional distributions, we initiate the study of distribution testing on structured multivariate distributions, and in particular the prototypical example of MRFs: the Ising Model. We demonstrate that, in this structured setting, we can avoid the curse of dimensionality, obtaining sample and time efficient testers for independence and goodness-of-fit. One of the key technical challenges we face along the way is bounding the variance of functions of the Ising model.