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We improve recently published results about resources of Restricted BoltzmannMachines (RBM) and Deep Belief Networks (DBN) required to make them UniversalApproximators. We show that any distribution p on the set of binary vectors oflength 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 oneentry such that their union contains the support set of p. In important casesthis number is half of the cardinality of the support set of p. We construct aDBN with 2^n/2(n-b), b ~ log(n), hidden layers of width n that is capable ofapproximating any distribution on {0,1}^n arbitrarily well. This confirms aconjecture presented by Le Roux and Bengio 2010.

Refinements of Universal Approximation Results for Deep Belief Networks and Restricted Boltzmann Machines