# Beyond Euclid: An Illustrated Guide to Modern Machine Learning with Geometric, Topological, and Algebraic Structures

arXiv: https://arxiv.org/abs/2407.09468 · HTML: https://mlsanity.com/paper/2407.09468/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- Builds on 133 of its 306 references that are in the map.
- First posted on arXiv 12 Jul 2024.

## Builds on (most cited first, 30 of 133)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [Neural Machine Translation by Jointly Learning to Align and Translate](https://mlsanity.com/paper/1409.0473/) (2014)
- [PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Space](https://mlsanity.com/paper/1706.02413/) (2017)
- [Efficient Estimation of Word Representations in Vector Space](https://mlsanity.com/paper/1301.3781/) (2013)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [node2vec: Scalable Feature Learning for Networks](https://mlsanity.com/paper/1607.00653/) (2016)
- [DeepWalk: Online Learning of Social Representations](https://mlsanity.com/paper/1403.6652/) (2014)
- [Point Transformer](https://mlsanity.com/paper/2011.00931/) (2020)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Variational Graph Auto-Encoders](https://mlsanity.com/paper/1611.07308/) (2016)
- [On the Continuity of Rotation Representations in Neural Networks](https://mlsanity.com/paper/1812.07035/) (2018)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Graph Convolutional Neural Networks for Web-Scale Recommender Systems](https://mlsanity.com/paper/1806.01973/) (2018)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Object-Centric Learning with Slot Attention](https://mlsanity.com/paper/2006.15055/) (2020)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [PCT: Point cloud transformer](https://mlsanity.com/paper/2012.09688/) (2020)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Equivariant Diffusion for Molecule Generation in 3D](https://mlsanity.com/paper/2203.17003/) (2022)

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

- [Position: Beyond Euclidean -- Foundation Models Should Embrace Non-Euclidean Geometries](https://mlsanity.com/paper/2504.08896/) (2025)
- [Flow Equivariant World Models: Memory for Partially Observed Dynamic Environments](https://mlsanity.com/paper/2601.01075/) (2026)
- [Wrapped Gaussian on the manifold of Symmetric Positive Definite Matrices](https://mlsanity.com/paper/2502.01512/) (2025)
- [go-mHC: Direct Parameterization of Manifold-Constrained Hyper-Connections via Generalized Orthostochastic Matrices](https://mlsanity.com/paper/2604.02309/) (2026)
- [Dynamical symmetries in the fluctuation-driven regime: an application of Noether's theorem to noisy dynamical systems](https://mlsanity.com/paper/2504.09761/) (2025)
- [Self-Organizing Visual Prototypes for Non-Parametric Representation Learning](https://mlsanity.com/paper/2505.21533/) (2025)
- [Intrinsic Dimensionality of Fermi-Pasta-Ulam-Tsingou High-Dimensional Trajectories Through Manifold Learning: A Linear Approach](https://mlsanity.com/paper/2411.02058/) (2024)
- [Busemann Functions in the Wasserstein Space: Existence, Closed-Forms, and Applications to Slicing](https://mlsanity.com/paper/2510.04579/) (2025)
- [TopoPrune: Robust Data Pruning via Unified Latent Space Topology](https://mlsanity.com/paper/2602.02739/) (2026)
- [A Mean Curvature Approach to Boundary Detection: Geometric Insights for Unsupervised Learning](https://mlsanity.com/paper/2605.04274/) (2026)
- [What Machine Learning Tells Us About the Mathematical Structure of Concepts](https://mlsanity.com/paper/2408.15507/) (2024)
- [Directed Semi-Simplicial Learning with Applications to Brain Activity Decoding](https://mlsanity.com/paper/2505.17939/) (2025)
- [Learning the Intrinsic Dimensionality of Fermi-Pasta-Ulam-Tsingou Trajectories: A Nonlinear Approach using a Deep Autoencoder Model](https://mlsanity.com/paper/2601.19567/) (2026)
- [Geometric Priors for Generalizable World Models via Vector Symbolic Architecture](https://mlsanity.com/paper/2602.21467/) (2026)
- [Curvature-Aware PCA with Geodesic Tangent Space Aggregation for Semi-Supervised Learning](https://mlsanity.com/paper/2604.18816/) (2026)
- [Riemannian geometry meets fMRI: the advantages of modeling correlation manifolds and eigenvector subspaces](https://mlsanity.com/paper/2605.22334/) (2026)
- [Efficient Mean Curvature Computation on High-Dimensional Data Manifolds](https://mlsanity.com/paper/2606.06329/) (2026)
- [CuBAS: Information Geometric Curvature-Based Adaptive Sampling for Supervised Classification](https://mlsanity.com/paper/2607.03145/) (2026)
- [Shape Operator PCA: Curvature-Aware Projections for Geometric Machine Learning](https://mlsanity.com/paper/2608.15313/) (2026)
- [Sample Complexity of Equivariant Reinforcement Learning](https://mlsanity.com/paper/2609.36421/) (2026)

## Abstract (arXiv)

The enduring legacy of Euclidean geometry underpins classical machine learning, which, for decades, has been primarily developed for data lying in Euclidean space. Yet, modern machine learning increasingly encounters richly structured data that is inherently nonEuclidean. This data can exhibit intricate geometric, topological and algebraic structure: from the geometry of the curvature of space-time, to topologically complex interactions between neurons in the brain, to the algebraic transformations describing symmetries of physical systems. Extracting knowledge from such non-Euclidean data necessitates a broader mathematical perspective. Echoing the 19th-century revolutions that gave rise to non-Euclidean geometry, an emerging line of research is redefining modern machine learning with non-Euclidean structures. Its goal: generalizing classical methods to unconventional data types with geometry, topology, and algebra. In this review, we provide an accessible gateway to this fast-growing field and propose a graphical taxonomy that integrates recent advances into an intuitive unified framework. We subsequently extract insights into current challenges and highlight exciting opportunities for future development in this field.