Quantitative Universal Approximation Bounds for Deep Belief Networks

Julian Sieber, Johann Gehringer

Keywords: Deep belief network, restricted Boltzmann machine, universal approximation property, expressivity, Kullback-Leibler approximation, probability density

Abstract: We show that deep belief networks with binary hidden units can approximate any multivariate probability density under very mild integrability requirements on the parental density of the visible nodes. The approximation is measured in the $L^q$-norm for $q\in[1,\infty]$ ($q=\infty$ corresponding to the supremum norm) and in Kullback-Leibler divergence. Furthermore, we establish sharp quantitative bounds on the approximation error in terms of the number of hidden units.

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TL;DR: We prove quantitative approximation results for deep belief networks with binary hidden units.