Pattern reconstruction with restricted Boltzmann machines

We show that the ability of a restricted Boltzmann machine to reconstruct a random pattern depends on the tail of the hidden prior distribution: hidden priors with strictly sub-Gaussian tails give only a logarithmic loss in pattern retrieval, while an efficient retrieval is much harder with hidden units with strictly super-Gaussian tails; reconstruction with sub-Gaussian hidden prior is regulated by the number of hidden units (as in the Hopfield model). This is proved by localisation estimates for the local minima of the energy function.