Contrastive divergence in gaussian diffusions

uous state version of recurrent neural networks. These networks are of interest for two reasons: (1) Powerful analytical tools exist that allow comparing the behavior of CD to other algorithms, like Maximum Likelihood estimation. (2) Many non-linear systems of interest for which CD has proven useful have multiple attractors about which the systems behave locally like GDs. Thus the analysis of the GD case may provide clues for a better understanding of CD in more general conditions. The analysis presented here shows that convergence of CD is guaranteed if the first moment of the GD is at equilibrium. In this case CD and maximum likelihood estimation converge to the same solution, otherwise CD may converge to arbitrarily different solutions from maximum likelihood estimation, or diverge altogether. In this paper we pursue a continuous-time formulation of CD that makes possible the use stochastic calculus tools. The continuous time case can be seen as the limit of the dynamics induced by the uncorrected discrete time Langevin Markov Chain Monte Carlo Method (Neal, 1996). In addition it should be noted that CD is typically interpreted as a method for learning equilibrium distributions while here we also examine it as a method for learning finite time distributions. Consider a stochastic process X = {Xt: t ∈ R+} defined by the following stochastic differential equation1 dXt = θ(γ −Xt)dt