  By controlling synaptic and neural correlations, deep learning has achieved
empirical successes in improving classification performances. How synaptic
correlations affect neural correlations to produce disentangled hidden
representations remains elusive. Here we propose a simplified model of
dimension reduction, taking into account pairwise correlations among synapses,
to reveal the mechanism underlying how the synaptic correlations affect
dimension reduction. Our theory determines the synaptic-correlation scaling
form requiring only mathematical self-consistency, for both binary and
continuous synapses. The theory also predicts that weakly-correlated synapses
encourage dimension reduction compared to their orthogonal counterparts. In
addition, these synapses slow down the decorrelation process along the network
depth. These two computational roles are explained by the proposed mean-field
equation. The theoretical predictions are in excellent agreement with numerical
simulations, and the key features are also captured by a deep learning with
Hebbian rules.
