# Hilbert geometry of the Siegel disk: The Siegel-Klein disk model

arXiv: https://arxiv.org/abs/2004.08160 · HTML: https://mlsanity.com/paper/2004.08160/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Apr 2020.

## Built on by (most cited first, 7 of 7)

- [A numerical approximation method for the Fisher-Rao distance between multivariate normal distributions](https://mlsanity.com/paper/2302.08175/) (2023)
- [Non-linear Embeddings in Hilbert Simplex Geometry](https://mlsanity.com/paper/2203.11434/) (2022)
- [Differential geometry with extreme eigenvalues in the positive semidefinite cone](https://mlsanity.com/paper/2304.07347/) (2023)
- [Hilbert geometry of the symmetric positive-definite bicone: Application to the geometry of the extended Gaussian family](https://mlsanity.com/paper/2508.14369/) (2025)
- [Towards the mathematical foundation of the minimum enclosing ball and related problems](https://mlsanity.com/paper/2402.06629/) (2024)
- [Siegel Neural Networks](https://mlsanity.com/paper/2511.09577/) (2025)
- [Flow matching on homogeneous spaces](https://mlsanity.com/paper/2603.24829/) (2026)

## Abstract (arXiv)

We study the Hilbert geometry induced by the Siegel disk domain, an open bounded convex set of complex square matrices of operator norm strictly less than one. This Hilbert geometry yields a generalization of the Klein disk model of hyperbolic geometry, henceforth called the Siegel-Klein disk model to differentiate it with the classical Siegel upper plane and disk domains. In the Siegel-Klein disk, geodesics are by construction always unique and Euclidean straight, allowing one to design efficient geometric algorithms and data-structures from computational geometry. For example, we show how to approximate the smallest enclosing ball of a set of complex square matrices in the Siegel disk domains: We compare two generalizations of the iterative core-set algorithm of Badoiu and Clarkson (BC) in the Siegel-Poincar'e disk and in the Siegel-Klein disk: We demonstrate that geometric computing in the Siegel-Klein disk allows one (i) to bypass the time-costly recentering operations to the disk origin required at each iteration of the BC algorithm in the Siegel-Poincar'e disk model, and (ii) to approximate fast and numerically the Siegel-Klein distance with guaranteed lower and upper bounds derived from nested Hilbert geometries.