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Modeling self-organization of neural networks for unsupervised learning usingHebbian and anti-Hebbian plasticity has a long history in neuroscience. Yet,derivations of single-layer networks with such local learning rules fromprincipled optimization objectives became possible only recently, with theintroduction of similarity matching objectives. What explains the success ofsimilarity matching objectives in deriving neural networks with local learningrules? Here, using dimensionality reduction as an example, we introduce severalvariable substitutions that illuminate the success of similarity matching. Weshow that the full network objective may be optimized separately for eachsynapse using local learning rules both in the offline and online settings. Weformalize the long-standing intuition of the rivalry between Hebbian andanti-Hebbian rules by formulating a min-max optimization problem. We introducea novel dimensionality reduction objective using fractional matrix exponents.To illustrate the generality of our approach, we apply it to a novelformulation of dimensionality reduction combined with whitening. We confirmnumerically that the networks with learning rules derived from principledobjectives perform better than those with heuristic learning rules.

Why do similarity matching objectives lead to Hebbian/anti-Hebbian networks?