  This work analyzes centered binary Restricted Boltzmann Machines (RBMs) and
binary Deep Boltzmann Machines (DBMs), where centering is done by subtracting
offset values from visible and hidden variables. We show analytically that (i)
centering results in a different but equivalent parameterization for artificial
neural networks in general, (ii) the expected performance of centered binary
RBMs/DBMs is invariant under simultaneous flip of data and offsets, for any
offset value in the range of zero to one, (iii) centering can be reformulated
as a different update rule for normal binary RBMs/DBMs, and (iv) using the
enhanced gradient is equivalent to setting the offset values to the average
over model and data mean. Furthermore, numerical simulations suggest that (i)
optimal generative performance is achieved by subtracting mean values from
visible as well as hidden variables, (ii) centered RBMs/DBMs reach
significantly higher log-likelihood values than normal binary RBMs/DBMs, (iii)
centering variants whose offsets depend on the model mean, like the enhanced
gradient, suffer from severe divergence problems, (iv) learning is stabilized
if an exponentially moving average over the batch means is used for the offset
values instead of the current batch mean, which also prevents the enhanced
gradient from diverging, (v) centered RBMs/DBMs reach higher LL values than
normal RBMs/DBMs while having a smaller norm of the weight matrix, (vi)
centering leads to an update direction that is closer to the natural gradient
and that the natural gradient is extremly efficient for training RBMs, (vii)
centering dispense the need for greedy layer-wise pre-training of DBMs, (viii)
furthermore we show that pre-training often even worsen the results
independently whether centering is used or not, and (ix) centering is also
beneficial for auto encoders.
