# Non-linear Embeddings in Hilbert Simplex Geometry

arXiv: https://arxiv.org/abs/2203.11434 · HTML: https://mlsanity.com/paper/2203.11434/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 48 references that are in the map.
- First posted on arXiv 22 Mar 2022.

## Builds on (most cited first, 13 of 13)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [DeepWalk: Online Learning of Social Representations](https://mlsanity.com/paper/1403.6652/) (2014)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Learning Continuous Hierarchies in the Lorentz Model of Hyperbolic Geometry](https://mlsanity.com/paper/1806.03417/) (2018)
- [Hyperbolic Image Embeddings](https://mlsanity.com/paper/1904.02239/) (2019)
- [Representation Tradeoffs for Hyperbolic Embeddings](https://mlsanity.com/paper/1804.03329/) (2018)
- [Learning the Predictability of the Future](https://mlsanity.com/paper/2101.01600/) (2021)
- [Tree! I am no Tree! I am a Low Dimensional Hyperbolic Embedding](https://mlsanity.com/paper/2005.03847/) (2020)
- [Symmetric Spaces for Graph Embeddings: A Finsler-Riemannian Approach](https://mlsanity.com/paper/2106.04941/) (2021)
- [Hilbert geometry of the Siegel disk: The Siegel-Klein disk model](https://mlsanity.com/paper/2004.08160/) (2020)
- [Embedding Node Structural Role Identity into Hyperbolic Space](https://mlsanity.com/paper/2011.01512/) (2020)
- [The Information-Geometric Perspective of Compositional Data Analysis](https://mlsanity.com/paper/2005.11510/) (2020)
- [Voronoi Diagrams in the Hilbert Metric](https://mlsanity.com/paper/2112.03056/) (2021)

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

- [Hilbert geometry of the symmetric positive-definite bicone: Application to the geometry of the extended Gaussian family](https://mlsanity.com/paper/2508.14369/) (2025)
- [The Tempered Hilbert Simplex Distance and Its Application To Non-linear Embeddings of TEMs](https://mlsanity.com/paper/2311.13459/) (2023)
- [Classifiers in High Dimensional Hilbert Metrics](https://mlsanity.com/paper/2601.13410/) (2026)
- [Geometric structures and deviations on James' symmetric positive-definite matrix bicone domain](https://mlsanity.com/paper/2603.02483/) (2026)
- [Toward a First-Principles Update Geometry for the Language-Model Head](https://mlsanity.com/paper/2608.22253/) (2026)
- [Taming Softmax at the Limits of Muon: A Unified Approach through Hilbert Geometry](https://mlsanity.com/paper/2610.17588/) (2026)

## Abstract (arXiv)

A key technique of machine learning and computer vision is to embed discrete weighted graphs into continuous spaces for further downstream processing. Embedding discrete hierarchical structures in hyperbolic geometry has proven very successful since it was shown that any weighted tree can be embedded in that geometry with arbitrary low distortion. Various optimization methods for hyperbolic embeddings based on common models of hyperbolic geometry have been studied. In this paper, we consider Hilbert geometry for the standard simplex which is isometric to a vector space equipped with the variation polytope norm. We study the representation power of this Hilbert simplex geometry by embedding distance matrices of graphs. Our findings demonstrate that Hilbert simplex geometry is competitive to alternative geometries such as the Poincar'e hyperbolic ball or the Euclidean geometry for embedding tasks while being fast and numerically robust.