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Ising models describe the joint probability distribution of a vector ofbinary feature variables. Typically, not all the variables interact with eachother and one is interested in learning the presumably sparse network structureof the interacting variables. However, in the presence of latent variables, theconventional method of learning a sparse model might fail. This is because thelatent variables induce indirect interactions of the observed variables. In thecase of only a few latent conditional Gaussian variables these spuriousinteractions contribute an additional low-rank component to the interactionparameters of the observed Ising model. Therefore, we propose to learn a sparse+ low-rank decomposition of the parameters of an Ising model using a convexregularized likelihood problem. We show that the same problem can be obtainedas the dual of a maximum-entropy problem with a new type of relaxation, wherethe sample means collectively need to match the expected values only up to agiven tolerance. The solution to the convex optimization problem hasconsistency properties in the high-dimensional setting, where the number ofobserved binary variables and the number of latent conditional Gaussianvariables are allowed to grow with the number of training samples.

Ising Models with Latent Conditional Gaussian Variables