MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV ( that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint

Deep Narrow Boltzmann Machines are Universal Approximators

Guido Montúfar


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. Besides from this existence statement, we provide upper and lower bounds on the sufficient number of layers and parameters. These bounds show that deep narrow Boltzmann machines are at least as compact universal approximators as restricted Boltzmann machines and narrow sigmoid belief networks, with respect to the currently available bounds for those models.


Related publications

2015 Repository Open Access
Guido Montúfar

Deep narrow Boltzmann machines are universal approximators

In: Third international conference on learning representations - ICLR 2015 : May 7-9 2015, San Diego, CA. USA
San Diego : ICLR, 2015.