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We show that deep narrow Boltzmann machines are universal approximators ofprobability distributions on the activities of their visible units, providedthey have sufficiently many hidden layers, each containing the same number ofunits as the visible layer. We show that, within certain parameter domains,deep Boltzmann machines can be studied as feedforward networks. We provideupper and lower bounds on the sufficient depth and width of universalapproximators. These results settle various intuitions regarding undirectednetworks and, in particular, they show that deep narrow Boltzmann machines areat least as compact universal approximators as narrow sigmoid belief networksand restricted Boltzmann machines, with respect to the currently availablebounds for those models.

Deep Narrow Boltzmann Machines are Universal Approximators