Refinements of universal approximation results for deep belief networks and restricted Boltzmann machines.

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 pon the set {0,1}(n) 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 the cardinality of the support set of p (given in Le Roux & Bengio, 2008). We construct a DBN with 2n/ 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 in Le Roux and Bengio (2010).