# Machine Learning Clifford invariants of ADE Coxeter elements

arXiv: https://arxiv.org/abs/2310.00041 · HTML: https://mlsanity.com/paper/2310.00041/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 78 references that are in the map.
- First posted on arXiv 29 Sep 2023.

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

- [Axiomatic Attribution for Deep Networks](https://mlsanity.com/paper/1703.01365/) (2017)
- [Learning to Unknot](https://mlsanity.com/paper/2010.16263/) (2020)
- [Machine Learning CICY Threefolds](https://mlsanity.com/paper/1806.03121/) (2018)
- [The Calabi-Yau Landscape: from Geometry, to Physics, to Machine-Learning](https://mlsanity.com/paper/1812.02893/) (2018)
- [Quiver Mutations, Seiberg Duality and Machine Learning](https://mlsanity.com/paper/2006.10783/) (2020)
- [Machine Learning Calabi-Yau Hypersurfaces](https://mlsanity.com/paper/2112.06350/) (2021)
- [Murmurations of elliptic curves](https://mlsanity.com/paper/2204.10140/) (2022)
- [Neurons on Amoebae](https://mlsanity.com/paper/2106.03695/) (2021)
- [Graded Symmetry Groups: Plane and Simple](https://mlsanity.com/paper/2107.03771/) (2021)
- [Machine-Learning Dessins d'Enfants: Explorations via Modular and Seiberg-Witten Curves](https://mlsanity.com/paper/2004.05218/) (2020)
- [Cluster Algebras: Network Science and Machine Learning](https://mlsanity.com/paper/2203.13847/) (2022)
- [Machine Learning Kreuzer--Skarke Calabi--Yau Threefolds](https://mlsanity.com/paper/2112.09117/) (2021)
- [Deep Learning Gauss-Manin Connections](https://mlsanity.com/paper/2007.13786/) (2020)
- [Machine Learning Algebraic Geometry for Physics](https://mlsanity.com/paper/2204.10334/) (2022)

## Built on by (most cited first, 6 of 6)

- [Machine Learning meets Algebraic Combinatorics: A Suite of Datasets Capturing Research-level Conjecturing Ability in Pure Mathematics](https://mlsanity.com/paper/2503.06366/) (2025)
- [Machine Learning Toric Duality in Brane Tilings](https://mlsanity.com/paper/2409.15251/) (2024)
- [Versor: A Geometric Sequence Architecture](https://mlsanity.com/paper/2602.10195/) (2026)
- [Learning 3-Manifold Triangulations](https://mlsanity.com/paper/2405.09610/) (2024)
- [Calabi-Yau Four/Five/Six-folds as mathbbPnw Hypersurfaces: Machine Learning, Approximation, and Generation](https://mlsanity.com/paper/2311.17146/) (2023)
- [FoGE: Fock Space inspired encoding for graph prompting](https://mlsanity.com/paper/2507.02937/) (2025)

## Abstract (arXiv)

There has been recent interest in novel Clifford geometric invariants of linear transformations. This motivates the investigation of such invariants for a certain type of geometric transformation of interest in the context of root systems, reflection groups, Lie groups and Lie algebras: the Coxeter transformations. We perform exhaustive calculations of all Coxeter transformations for A8, D8 and E8 for a choice of basis of simple roots and compute their invariants, using high-performance computing. This computational algebra paradigm generates a dataset that can then be mined using techniques from data science such as supervised and unsupervised machine learning. In this paper we focus on neural network classification and principal component analysis. Since the output -- the invariants -- is fully determined by the choice of simple roots and the permutation order of the corresponding reflections in the Coxeter element, we expect huge degeneracy in the mapping. This provides the perfect setup for machine learning, and indeed we see that the datasets can be machine learned to very high accuracy. This paper is a pump-priming study in experimental mathematics using Clifford algebras, showing that such Clifford algebraic datasets are amenable to machine learning, and shedding light on relationships between these novel and other well-known geometric invariants and also giving rise to analytic results.