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We study the identity testing problem for restricted Boltzmann machines (RBMs), and more generally, for undirected graphical models. In this problem, given sample access to the Gibbs distribution corresponding to an unknown or hidden model M* and given an explicit model M, the goal is to distinguish if either M = M* or if the models are (statistically) far apart. We establish the computational hardness of identity testing for RBMs (i.e., mixed Ising models on bipartite graphs), even when there are no latent variables or an external field. Specifically, we show that unless RP = NP, there is no polynomial-time identity testing algorithm for RBMs when fid = !(log n), where d is the maximum degree of the visible graph and fi is the largest edge weight (in absolute value); when fid = O(log n) there is an efficient identity testing algorithm that utilizes the structure learning algorithm of Klivans and Meka (2017). We prove similar lower bounds for purely ferromagnetic RBMs with inconsistent external fields and for the ferromagnetic Potts model. To prove our results, we introduce a novel methodology to reduce the corresponding approximate counting problem to testing utilizing the phase transition exhibited by these models. [ABSTRACT FROM AUTHOR]

We study the identity testing problem in the context of spin systems or undirected graphical models, where it takes the following form: given the parameter specification of the model M and a sampling oracle for the distribution μM* of an unknown model M*, can we efficiently determine if the two models M and M* are the same? We consider identity testing for both soft-constraint and hard-constraint systems. In particular, we prove hardness results in two prototypical cases, the Ising model and proper colorings, and explore whether identity testing is any easier than structure learning. For the ferromagnetic (attractive) Ising model, Daskalakis et al. (2018) presented a polynomial time algorithm for identity testing. We prove hardness results in the antiferromagnetic (repulsive) setting in the same regime of parameters where structure learning is known to require a super-polynomial number of samples. Specifically, for n-vertex graphs of maximum degree d, we prove that if |β|d = ω(log n) (where β is the inverse temperature parameter), then there is no polynomial running time identity testing algorithm unless RP = NP. In the hard-constraint setting, we present hardness results for identity testing for proper colorings. Our results are based on the presumed hardness of #BIS, the problem of (approximately) counting independent sets in bipartite graphs. [ABSTRACT FROM AUTHOR]