# K"ahler Geometry of Quiver Varieties and Machine Learning

arXiv: https://arxiv.org/abs/2101.11487 · HTML: https://mlsanity.com/paper/2101.11487/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 58 references that are in the map.
- First posted on arXiv 27 Jan 2021.

## Builds on (most cited first, 14 of 14)

- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [Hyperbolic Neural Networks](https://mlsanity.com/paper/1805.09112/) (2018)
- [Hyperbolic Graph Convolutional Neural Networks](https://mlsanity.com/paper/1910.12933/) (2019)
- [The Expressive Power of Neural Networks: A View from the Width](https://mlsanity.com/paper/1709.02540/) (2017)
- [Identifying and attacking the saddle point problem in high-dimensional non-convex optimization](https://mlsanity.com/paper/1406.2572/) (2014)
- [Gauge Equivariant Convolutional Networks and the Icosahedral CNN](https://mlsanity.com/paper/1902.04615/) (2019)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [ResNet with one-neuron hidden layers is a Universal Approximator](https://mlsanity.com/paper/1806.10909/) (2018)
- [Natural Graph Networks](https://mlsanity.com/paper/2007.08349/) (2020)
- [Learning Activation Functions: A new paradigm for understanding Neural Networks](https://mlsanity.com/paper/1906.09529/) (2019)
- [On the saddle point problem for non-convex optimization](https://mlsanity.com/paper/1405.4604/) (2014)
- [The Representation Theory of Neural Networks](https://mlsanity.com/paper/2007.12213/) (2020)
- [Graph Laplacians, Riemannian Manifolds and their Machine-Learning](https://mlsanity.com/paper/2006.16619/) (2020)

## Built on by (most cited first, 3 of 3)

- [Quiver neural networks](https://mlsanity.com/paper/2207.12773/) (2022)
- [Noncommutative Geometry of Computational Models and Uniformization for Framed Quiver Varieties](https://mlsanity.com/paper/2201.05900/) (2022)
- [Quantum Finite Automata and Quiver Algebras](https://mlsanity.com/paper/2203.07597/) (2022)

## Abstract (arXiv)

We develop an algebro-geometric formulation for neural networks in machine learning using the moduli space of framed quiver representations. We find natural Hermitian metrics on the universal bundles over the moduli which are compatible with the GIT quotient construction by the general linear group, and show that their Ricci curvatures give a K"ahler metric on the moduli. Moreover, we use toric moment maps to construct activation functions, and prove the universal approximation theorem for the multi-variable activation function constructed from the complex projective space.