  Restricted Boltzmann machines are energy models made of a visible and a
hidden layer. We identify an effective energy function describing the
zero-temperature landscape on the visible units and depending only on the tail
behaviour of the hidden layer prior distribution. Studying the location of the
local minima of such an energy function, we show that the ability of a
restricted Boltzmann machine to reconstruct a random pattern depends indeed
only on the tail of the hidden prior distribution. We find that hidden priors
with strictly super-Gaussian tails give only a logarithmic loss in pattern
retrieval, while an efficient retrieval is much harder with hidden units with
strictly sub-Gaussian tails; if the hidden prior has Gaussian tails, the
retrieval capability is determined by the number of hidden units (as in the
Hopfield model).
