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Graphical models are powerful tools for modeling high-dimensional data, butlearning graphical models in the presence of latent variables is well-known tobe difficult. In this work we give new results for learning RestrictedBoltzmann Machines, probably the most well-studied class of latent variablemodels. Our results are based on new connections to learning two-layer neuralnetworks under $\ell_{\infty}$ bounded input; for both problems, we give nearlyoptimal results under the conjectured hardness of sparse parity with noise.Using the connection between RBMs and feedforward networks, we also initiatethe theoretical study of $supervised~RBMs$ [Hinton, 2012], a version ofneural-network learning that couples distributional assumptions induced fromthe underlying graphical model with the architecture of the unknown functionclass. We then give an algorithm for learning a natural class of supervisedRBMs with better runtime than what is possible for its related class ofnetworks without distributional assumptions.

From Boltzmann Machines to Neural Networks and Back Again