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

arXiv: https://arxiv.org/abs/1703.07914 · HTML: https://mlsanity.com/paper/1703.07914/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Mar 2017.

## Built on by (most cited first, 20 of 20)

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- [Kernel similarity matching with Hebbian neural networks](https://mlsanity.com/paper/2204.07475/) (2022)
- [Unlocking the Potential of Similarity Matching: Scalability, Supervision and Pre-training](https://mlsanity.com/paper/2308.02427/) (2023)
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- [Ubiquity of Emergent Hebbian Dynamics in Regularized Learning](https://mlsanity.com/paper/2505.18069/) (2025)
- [Similarity Matching Networks: Hebbian Learning and Convergence Over Multiple Time Scales](https://mlsanity.com/paper/2506.06134/) (2025)
- [Rethinking Hebbian Principle: Low-Dimensional Structural Projection for Unsupervised Learning](https://mlsanity.com/paper/2510.14810/) (2025)

## Abstract (arXiv)

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.