# mathbbSL(n) Representation Learning: An Intrinsic Mixed-Curvature Space with Higher Curvature Capacities and Deeper Order-Aware Composition

arXiv: https://arxiv.org/abs/2609.15083 · HTML: https://mlsanity.com/paper/2609.15083/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 24 of its 43 references that are in the map.
- First posted on arXiv 14 Sep 2026.

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

- [Distributed Representations of Words and Phrases and their Compositionality](https://mlsanity.com/paper/1310.4546/) (2013)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Hyperbolic Graph Convolutional Neural Networks](https://mlsanity.com/paper/1910.12933/) (2019)
- [Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data](https://mlsanity.com/paper/2002.12880/) (2020)
- [Low-Dimensional Hyperbolic Knowledge Graph Embeddings](https://mlsanity.com/paper/2005.00545/) (2020)
- [Constant Curvature Graph Convolutional Networks](https://mlsanity.com/paper/1911.05076/) (2019)
- [LieTransformer: Equivariant self-attention for Lie Groups](https://mlsanity.com/paper/2012.10885/) (2020)
- [Mixed-curvature Variational Autoencoders](https://mlsanity.com/paper/1911.08411/) (2019)
- [Automatic Symmetry Discovery with Lie Algebra Convolutional Network](https://mlsanity.com/paper/2109.07103/) (2021)
- [Building Deep Networks on Grassmann Manifolds](https://mlsanity.com/paper/1611.05742/) (2016)
- [Deep Learning on Lie Groups for Skeleton-based Action Recognition](https://mlsanity.com/paper/1612.05877/) (2016)
- [A Self-supervised Mixed-curvature Graph Neural Network](https://mlsanity.com/paper/2112.05393/) (2021)
- [GraphMoRE: Mitigating Topological Heterogeneity via Mixture of Riemannian Experts](https://mlsanity.com/paper/2412.11085/) (2024)
- [Symmetric Spaces for Graph Embeddings: A Finsler-Riemannian Approach](https://mlsanity.com/paper/2106.04941/) (2021)
- [Lie Group Decompositions for Equivariant Neural Networks](https://mlsanity.com/paper/2310.11366/) (2023)
- [Modeling Graphs Beyond Hyperbolic: Graph Neural Networks in Symmetric Positive Definite Matrices](https://mlsanity.com/paper/2306.14064/) (2023)
- [Finsler Multi-Dimensional Scaling: Manifold Learning for Asymmetric Dimensionality Reduction and Embedding](https://mlsanity.com/paper/2503.18010/) (2025)
- [Lie Neurons: Adjoint-Equivariant Neural Networks for Semisimple Lie Algebras](https://mlsanity.com/paper/2310.04521/) (2023)
- [Improving Heterogeneous Graph Learning with Weighted Mixed-Curvature Product Manifold](https://mlsanity.com/paper/2307.04514/) (2023)
- [Product Manifold Representations for Learning on Biological Pathways](https://mlsanity.com/paper/2401.15478/) (2024)
- [Equivariant Neural Networks for General Linear Symmetries on Lie Algebras](https://mlsanity.com/paper/2510.22984/) (2025)
- [Learning Polynomial Problems with SL(2,ℝ) Equivariance](https://mlsanity.com/paper/2312.02146/) (2023)
- [Finsler Geometry, Graph Neural Networks, and You](https://mlsanity.com/paper/2606.17185/) (2026)

## Abstract (arXiv)

Mixed-curvature representation learning seeks to capture rich geometric structures that cannot be adequately modeled by a single curvature regime. Existing approaches largely rely on product manifolds, which require manually specifying how different curvature spaces are combined and separate their curvature contributions across factors. We introduce the mathbbSL(n) space, a representation geometry defined by the simple det(A)=1 constraint and a left invariant Schatten-p Finsler structure. Despite this minimal construction, mathbbSL(n) exhibits pointwise negative, zero, and positive flag curvature around a common flagpole, while its mixed-curvature and curvature-coupling capacities are asymptotically maximal relative to the intrinsic geometric upper bound. Beyond geometry, its noncommutative group structure provides inherent order sensitivity, and its non-nilpotent Lie algebra admits nonzero nested Lie brackets at arbitrary depth, enabling deep order-aware composition. Empirically, mathbbSL(n) consistently outperforms a broad range of representation manifold baselines across graph benchmarks at different scales. It reduces average distortion over the strongest baselines by 44.3% on KEGG and 40.5% on HumanCyc, and improves Hits@20 by 42.8% on OGBL-PPA. Experiments on Flickr30k-Order further support its ability to capture higher order dependencies from ordered composition. Together, these results show how a seemingly simple structural constraint can yield unexpectedly rich geometry, capacity, and composition within a unified representation space.