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

Expressive Power and Approximation Errors of Restricted Boltzmann Machines

Guido Montúfar, Johannes Rauh and Nihat Ay


We present explicit classes of probability distributions that can be learned by Restricted Boltzmann Machines (RBMs) depending on the number of units that they contain, and which are representative for the expressive power of the model. We use this to show that the maximal Kullback-Leibler divergence to the RBM model with $n$ visible and $m$ hidden units is bounded from above by $(n-1)-\log(m+1)$.

In this way we can specify the number of hidden units that guarantees a sufficiently rich model containing different classes of distributions and respecting a given error tolerance.

MSC Codes:
68Q32, 68T01, 62-04
Machine Learning, neural networks, Unsupervised Learning, Representational Power, Universal approximator

Related publications

2011 Repository Open Access
Guido Montúfar, Johannes Rauh and Nihat Ay

Expressive power and approximation errors of restricted Boltzmann machines

In: Advances in neural information processing systems 24 : NIPS 2011 ; 25th annual conference on neural information processing systems 2011, Granada, Spain December 12th - 15th / John Shawe-Taylor (ed.)
La Jolla, CA : Neural Information Processing Systems, 2011. - pp. 415-423