We improve recently published results about resources of restricted Boltzmann machines (RBM) and deep belief networks (DBN) required to make them universal approximators. We show that any distribution on the set of binary vectors of length can be arbitrarily well approximated by an RBM with hidden units, where is the minimal number of pairs of binary vectors differing in only one entry such that their union contains the support set of . In important cases this number is half the cardinality of the support set of (given in Le Roux & Bengio, 2008 ). We construct a DBN with , hidden layers of width that is capable of approximating any distribution on arbitrarily well. This confirms a conjecture presented in Le Roux and Bengio ( 2010 ).

We extend Linkser's Infomax principle for feedforward neural networks to a measure for stochastic interdependence that captures spatial and temporal signal properties in recurrent systems. This measure, stochastic interaction , quantifies the Kullback-Leibler divergence of a Markov chain from a product of split chains for the single unit processes. For unconstrained Markov chains, the maximization of stochastic interaction, also called Temporal Infomax , has been previously shown to result in almost deterministic dynamics. This letter considers Temporal Infomax on constrained Markov chains, where some of the units are clamped to prescribed stochastic processes providing input to the system. Temporal Infomax in that case leads to finite state automata, either completely deterministic or weakly nondeterministic. Transitions between internal states of these systems are almost perfectly predictable given the complete current state and the input, but the activity of each single unit alone is virtually random. The results are demonstrated by means of computer simulations and confirmed analytically. It is furthermore shown numerically that Temporal Infomax leads to a high information flow from the input to internal units and that a simple temporal learning rule can approximately achieve the optimization of temporal interaction. We relate these results to experimental data concerning the correlation dynamics and functional connectivities observed in multiple electrode recordings.

The hypothesis of invariant maximization of interaction (IMI) is formulated within the setting of random fields. According to this hypothesis, learning processes maximize the stochastic interaction of the neurons subject to constraints. We consider the extrinsic constraint in terms of a fixed input distribution on the periphery of the network. Our main intrinsic constraint is given by a directed acyclic network structure. First mathematical results about the strong relation of the local information flow and the global interaction are stated in order to investigate the possibility of controlling IMI optimization in a completely local way. Furthermore, we discuss some relations of this approach to the optimization according to Linsker's Infomax principle.