Guido Montufar
Mixture Decomposition of Distributions using a Decomposition of the Sample Space.
Tue, 08/03/2010 - 15:01 — SomNambulAuthors: Guido Montufar
Subjects: Statistics
AbstractWe consider the set of join probability distributions of $N$ binary random
variables which can be written as a sum of $m$ distributions in the following
form $p(x_1,\ldots,x_N)=\sum_{i=1}^m \alpha_i f_i(x_1,\ldots,x_N)$, where
$\alpha_i \geq 0$, $\sum_{i=1}^m \alpha_i =1$, and the $f_i(x_1,\ldots,x_N)$
belong to some exponential family. For our analysis we decompose the sample
space into portions on which the mixture components $f_i$ can be chosen
arbitrarily.Refinements of Universal Approximation Results for Deep Belief Networks and Restricted Boltzmann Machines.
Tue, 05/11/2010 - 15:01 — SomNambulAuthors: Nihat Ay, Guido Montufar
Subjects: Machine Learning
AbstractWe 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.
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