  We consider deep multi-layered generative models such as Boltzmann machines
or Hopfield nets in which computation (which implements inference) is both
recurrent and stochastic, but where the recurrence is not to model sequential
structure, only to perform computation. We find conditions under which a simple
feedforward computation is a very good initialization for inference, after the
input units are clamped to observed values. It means that after the feedforward
initialization, the recurrent network is very close to a fixed point of the
network dynamics, where the energy gradient is 0. The main condition is that
consecutive layers form a good auto-encoder, or more generally that different
groups of inputs into the unit (in particular, bottom-up inputs on one hand,
top-down inputs on the other hand) are consistent with each other, producing
the same contribution into the total weighted sum of inputs. In biological
terms, this would correspond to having each dendritic branch correctly
predicting the aggregate input from all the dendritic branches, i.e., the soma
potential. This is consistent with the prediction that the synaptic weights
into dendritic branches such as those of the apical and basal dendrites of
pyramidal cells are trained to minimize the prediction error made by the
dendritic branch when the target is the somatic activity. Whereas previous work
has shown how to achieve fast negative phase inference (when the model is
unclamped) in a predictive recurrent model, this contribution helps to achieve
fast positive phase inference (when the target output is clamped) in such
recurrent neural models.
