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