  Modeling self-organization of neural networks for unsupervised learning using
Hebbian and anti-Hebbian plasticity has a long history in neuroscience. Yet,
derivations of single-layer networks with such local learning rules from
principled optimization objectives became possible only recently, with the
introduction of similarity matching objectives. What explains the success of
similarity matching objectives in deriving neural networks with local learning
rules? Here, using dimensionality reduction as an example, we introduce several
variable substitutions that illuminate the success of similarity matching. We
show that the full network objective may be optimized separately for each
synapse using local learning rules both in the offline and online settings. We
formalize the long-standing intuition of the rivalry between Hebbian and
anti-Hebbian rules by formulating a min-max optimization problem. We introduce
a novel dimensionality reduction objective using fractional matrix exponents.
To illustrate the generality of our approach, we apply it to a novel
formulation of dimensionality reduction combined with whitening. We confirm
numerically that the networks with learning rules derived from principled
objectives perform better than those with heuristic learning rules.
