Expressive Power of Conditional Restricted Boltzmann Machines

Conditional restricted Boltzmann machines are undirected stochastic neural networks with a layer of input and output units connected bipartitely to a layer of hidden units. These networks define models of conditional probability distributions on the states of the output units given the states of the input units, parametrized by interaction weights and biases. We address the representational power of these models, proving results on the minimal size of universal approximators of conditional probability distributions, the minimal size of universal approximators of deterministic functions, the maximal model approximation errors, and on the dimension of the set of representable conditional distributions. We contribute new tools for investigating conditional models and obtain significant improvements over the results that can be derived directly from existing work on restricted Boltzmann machine probability models.