# The Representation Theory of Neural Networks

arXiv: https://arxiv.org/abs/2007.12213 · HTML: https://mlsanity.com/paper/2007.12213/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Jul 2020.

## Built on by (most cited first, 19 of 19)

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- [Symmetry Teleportation for Accelerated Optimization](https://mlsanity.com/paper/2205.10637/) (2022)
- [Improving Convergence and Generalization Using Parameter Symmetries](https://mlsanity.com/paper/2305.13404/) (2023)
- [Beyond the Permutation Symmetry of Transformers: The Role of Rotation for Model Fusion](https://mlsanity.com/paper/2502.00264/) (2025)
- [TeleSparse: Practical Privacy-Preserving Verification of Deep Neural Networks](https://mlsanity.com/paper/2504.19274/) (2025)
- [Universal approximation and model compression for radial neural networks](https://mlsanity.com/paper/2107.02550/) (2021)
- [Topology and Geometry of the Learning Space of ReLU Networks: Connectivity and Singularities](https://mlsanity.com/paper/2602.00693/) (2026)
- [Quiver neural networks](https://mlsanity.com/paper/2207.12773/) (2022)
- [K"ahler Geometry of Quiver Varieties and Machine Learning](https://mlsanity.com/paper/2101.11487/) (2021)
- [Noncommutative Geometry of Computational Models and Uniformization for Framed Quiver Varieties](https://mlsanity.com/paper/2201.05900/) (2022)
- [A logifold structure on measure space](https://mlsanity.com/paper/2405.05492/) (2024)
- [Teleportation With Null Space Gradient Projection for Optimization Acceleration](https://mlsanity.com/paper/2502.11362/) (2025)
- [The Epistemological Consequences of Large Language Models: Rethinking collective intelligence and institutional knowledge](https://mlsanity.com/paper/2512.19570/) (2025)
- [Level Set Teleportation: An Optimization Perspective](https://mlsanity.com/paper/2403.03362/) (2024)
- [Hidden Activations Are Not Enough: A General Approach to Neural Network Predictions](https://mlsanity.com/paper/2409.13163/) (2024)
- [Quantum Finite Automata and Quiver Algebras](https://mlsanity.com/paper/2203.07597/) (2022)
- [Advancing Neural Network Performance through Emergence-Promoting Initialization Scheme](https://mlsanity.com/paper/2407.19044/) (2024)
- [PAGR: Proof-Carrying Algebraic-Geometric Retrieval: A Quiver-, Provenance-, and Sheaf-Theoretic Framework for Grounded LLM Retrieval](https://mlsanity.com/paper/2609.06127/) (2026)
- [Hidden Activations are not Enough I: Knowledge Matrices as Higher Representations](https://mlsanity.com/paper/2609.34166/) (2026)

## Abstract (arXiv)

In this work, we show that neural networks can be represented via the mathematical theory of quiver representations. More specifically, we prove that a neural network is a quiver representation with activation functions, a mathematical object that we represent using a network quiver. Also, we show that network quivers gently adapt to common neural network concepts such as fully-connected layers, convolution operations, residual connections, batch normalization, pooling operations and even randomly wired neural networks. We show that this mathematical representation is by no means an approximation of what neural networks are as it exactly matches reality. This interpretation is algebraic and can be studied with algebraic methods. We also provide a quiver representation model to understand how a neural network creates representations from the data. We show that a neural network saves the data as quiver representations, and maps it to a geometrical space called the moduli space, which is given in terms of the underlying oriented graph of the network, i.e., its quiver. This results as a consequence of our defined objects and of understanding how the neural network computes a prediction in a combinatorial and algebraic way. Overall, representing neural networks through the quiver representation theory leads to 9 consequences and 4 inquiries for future research that we believe are of great interest to better understand what neural networks are and how they work.