  Synaptic plasticity is widely accepted to be the mechanism behind learning in
the brain's neural networks. A central question is how synapses, with access to
only local information about the network, can still organize collectively and
perform circuit-wide learning in an efficient manner. In single-layered and
all-to-all connected neural networks, local plasticity has been shown to
implement gradient-based learning on a class of cost functions that contain a
term that aligns the similarity of outputs to the similarity of inputs. Whether
such cost functions exist for networks with other architectures is not known.
In this paper, we introduce structured and deep similarity matching cost
functions, and show how they can be optimized in a gradient-based manner by
neural networks with local learning rules. These networks extend F\"oldiak's
Hebbian/Anti-Hebbian network to deep architectures and structured feedforward,
lateral and feedback connections. Credit assignment problem is solved elegantly
by a factorization of the dual learning objective to synapse specific local
objectives. Simulations show that our networks learn meaningful features.
