  Unsupervised learning requiring only raw data is not only a fundamental
function of the cerebral cortex, but also a foundation for a next generation of
artificial neural networks. However, a unified theoretical framework to treat
sensory inputs, synapses and neural activity together is still lacking. The
computational obstacle originates from the discrete nature of synapses, and
complex interactions among these three essential elements of learning. Here, we
propose a variational mean-field theory in which the distribution of synaptic
weights is considered. The unsupervised learning can then be decomposed into
two intertwined steps: a maximization step is carried out as a gradient ascent
of the lower-bound on the data log-likelihood, in which the synaptic weight
distribution is determined by updating variational parameters, and an
expectation step is carried out as a message passing procedure on an equivalent
or dual neural network whose parameter is specified by the variational
parameters of the weight distribution. Therefore, our framework provides
insights on how data (or sensory inputs), synapses and neural activities
interact with each other to achieve the goal of extracting statistical
regularities in sensory inputs. This variational framework is verified in
restricted Boltzmann machines with planted synaptic weights and
handwritten-digits learning.
