  Olshausen and Field (OF) proposed that neural computations in the primary
visual cortex (V1) can be partially modeled by sparse dictionary learning. By
minimizing the regularized representation error they derived an online
algorithm, which learns Gabor-filter receptive fields from a natural image
ensemble in agreement with physiological experiments. Whereas the OF algorithm
can be mapped onto the dynamics and synaptic plasticity in a single-layer
neural network, the derived learning rule is nonlocal - the synaptic weight
update depends on the activity of neurons other than just pre- and postsynaptic
ones - and hence biologically implausible. Here, to overcome this problem, we
derive sparse dictionary learning from a novel cost-function - a regularized
error of the symmetric factorization of the input's similarity matrix. Our
algorithm maps onto a neural network of the same architecture as OF but using
only biologically plausible local learning rules. When trained on natural
images our network learns Gabor-filter receptive fields and reproduces the
correlation among synaptic weights hard-wired in the OF network. Therefore,
online symmetric matrix factorization may serve as an algorithmic theory of
neural computation.
