Deep Narrow Boltzmann Machines are Universal Approximators
This provides theoretical justification for the compactness of deep narrow Boltzmann machines compared to other models like sigmoid belief networks and restricted Boltzmann machines, addressing foundational questions in undirected network theory.
The paper demonstrates that deep narrow Boltzmann machines can universally approximate probability distributions on visible units given enough hidden layers of equal width, establishing upper and lower bounds on depth and width requirements.
We show that deep narrow Boltzmann machines are universal approximators of probability distributions on the activities of their visible units, provided they have sufficiently many hidden layers, each containing the same number of units as the visible layer. We show that, within certain parameter domains, deep Boltzmann machines can be studied as feedforward networks. We provide upper and lower bounds on the sufficient depth and width of universal approximators. These results settle various intuitions regarding undirected networks and, in particular, they show that deep narrow Boltzmann machines are at least as compact universal approximators as narrow sigmoid belief networks and restricted Boltzmann machines, with respect to the currently available bounds for those models.