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

## TL;DR

This paper explains why similarity matching objectives naturally lead to neural networks with Hebbian and anti-Hebbian learning rules, using dimensionality reduction as a case study and introducing novel optimization formulations.

## Contribution

It introduces a formal framework connecting similarity matching objectives to local learning rules and demonstrates improved network performance over heuristic methods.

## Key findings

- Networks derived from principled objectives outperform heuristic ones.
- A novel dimensionality reduction objective with fractional matrix exponents is proposed.
- The approach generalizes to combined dimensionality reduction and whitening.

## Abstract

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.