Deep Belief Networks Are Compact Universal Approximators

Deep Belief Networks (DBN) are generative models with many layers of hidden causal variables, recently introduced by Hinton et al. (2006), along with a greedy layer-wise unsupervised learning algorithm. Building on Le Roux and Bengio (2008) and Sutskever and Hinton (2008), we show that deep but narrow generative networks do not require more parameters than shallow ones to achieve universal approximation. Exploiting the proof technique, we prove that deep but narrow feed-forward neural networks with sigmoidal units can represent any Boolean expression.