  We improve recently published results about resources of Restricted Boltzmann
Machines (RBM) and Deep Belief Networks (DBN) required to make them Universal
Approximators. We show that any distribution p on the set of binary vectors of
length n can be arbitrarily well approximated by an RBM with k-1 hidden units,
where k is the minimal number of pairs of binary vectors differing in only one
entry such that their union contains the support set of p. In important cases
this number is half of the cardinality of the support set of p. We construct a
DBN with 2^n/2(n-b), b ~ log(n), hidden layers of width n that is capable of
approximating any distribution on {0,1}^n arbitrarily well. This confirms a
conjecture presented by Le Roux and Bengio 2010.
