# If you can distinguish, you can express: Galois theory, Stone--Weierstrass, machine learning, and linguistics

arXiv: https://arxiv.org/abs/2510.09902 · HTML: https://mlsanity.com/paper/2510.09902/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 24 of its 91 references that are in the map.
- First posted on arXiv 10 Oct 2025.

## Builds on (most cited first, 24 of 24)

- [ShapeNet: An Information-Rich 3D Model Repository](https://mlsanity.com/paper/1512.03012/) (2015)
- [Convolutional Networks on Graphs for Learning Molecular Fingerprints](https://mlsanity.com/paper/1509.09292/) (2015)
- [Benign Overfitting in Linear Regression](https://mlsanity.com/paper/1906.11300/) (2019)
- [Universal Invariant and Equivariant Graph Neural Networks](https://mlsanity.com/paper/1905.04943/) (2019)
- [On the equivalence between graph isomorphism testing and function approximation with GNNs](https://mlsanity.com/paper/1905.12560/) (2019)
- [Frame Averaging for Invariant and Equivariant Network Design](https://mlsanity.com/paper/2110.03336/) (2021)
- [On the Universality of Invariant Networks](https://mlsanity.com/paper/1901.09342/) (2019)
- [Universal approximations of invariant maps by neural networks](https://mlsanity.com/paper/1804.10306/) (2018)
- [Weisfeiler and Leman go Machine Learning: The Story so far](https://mlsanity.com/paper/2112.09992/) (2021)
- [Equivariance with Learned Canonicalization Functions](https://mlsanity.com/paper/2211.06489/) (2022)
- [On the Universality of Rotation Equivariant Point Cloud Networks](https://mlsanity.com/paper/2010.02449/) (2020)
- [Universal Equivariant Multilayer Perceptrons](https://mlsanity.com/paper/2002.02912/) (2020)
- [Low Dimensional Invariant Embeddings for Universal Geometric Learning](https://mlsanity.com/paper/2205.02956/) (2022)
- [Fine-grained Expressivity of Graph Neural Networks](https://mlsanity.com/paper/2306.03698/) (2023)
- [Group-invariant max filtering](https://mlsanity.com/paper/2205.14039/) (2022)
- [Estimation under group actions: recovering orbits from invariants](https://mlsanity.com/paper/1712.10163/) (2017)
- [Towards fully covariant machine learning](https://mlsanity.com/paper/2301.13724/) (2023)
- [Maximum likelihood for high-noise group orbit estimation and single-particle cryo-EM](https://mlsanity.com/paper/2107.01305/) (2021)
- [The generalized method of moments for multi-reference alignment](https://mlsanity.com/paper/2103.02215/) (2021)
- [Max filtering with reflection groups](https://mlsanity.com/paper/2212.05104/) (2022)
- [On Universality Classes of Equivariant Networks](https://mlsanity.com/paper/2506.02293/) (2025)
- [Dihedral multi-reference alignment](https://mlsanity.com/paper/2107.05262/) (2021)
- [Sparse multi-reference alignment: sample complexity and computational hardness](https://mlsanity.com/paper/2109.11656/) (2021)
- [The generalized phase retrieval problem over compact groups](https://mlsanity.com/paper/2501.03549/) (2025)

## Abstract (arXiv)

This essay develops a parallel between the Fundamental Theorem of Galois Theory and the Stone--Weierstrass theorem: both can be viewed as assertions that tie the distinguishing power of a class of objects to their expressive power. We provide an elementary theorem connecting the relevant notions of "distinguishing power". We also discuss machine learning and data science contexts in which these theorems, and more generally the theme of links between distinguishing power and expressive power, appear. Finally, we discuss the same theme in the context of linguistics, where it appears as a foundational principle, and illustrate it with several examples.