# Differential geometry with extreme eigenvalues in the positive semidefinite cone

arXiv: https://arxiv.org/abs/2304.07347 · HTML: https://mlsanity.com/paper/2304.07347/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 68 references that are in the map.
- First posted on arXiv 14 Apr 2023.

## Builds on (most cited first, 18 of 18)

- [A Riemannian Network for SPD Matrix Learning](https://mlsanity.com/paper/1608.04233/) (2016)
- [Manopt, a Matlab toolbox for optimization on manifolds](https://mlsanity.com/paper/1308.5200/) (2013)
- [Riemannian batch normalization for SPD neural networks](https://mlsanity.com/paper/1909.02414/) (2019)
- [geomstats: a Python Package for Riemannian Geometry in Machine Learning](https://mlsanity.com/paper/1805.08308/) (2018)
- [Kernel Methods on Riemannian Manifolds with Gaussian RBF Kernels](https://mlsanity.com/paper/1412.0265/) (2014)
- [Riemannian Dictionary Learning and Sparse Coding for Positive Definite Matrices](https://mlsanity.com/paper/1507.02772/) (2015)
- [Geometric Mean Metric Learning](https://mlsanity.com/paper/1607.05002/) (2016)
- [Tensor-CSPNet: A Novel Geometric Deep Learning Framework for Motor Imagery Classification](https://mlsanity.com/paper/2202.02472/) (2022)
- [Clustering Signed Networks with the Geometric Mean of Laplacians](https://mlsanity.com/paper/1701.00757/) (2017)
- [Efficient Algorithms for Large-scale Generalized Eigenvector Computation and Canonical Correlation Analysis](https://mlsanity.com/paper/1604.03930/) (2016)
- [Geometry-aware Similarity Learning on SPD Manifolds for Visual Recognition](https://mlsanity.com/paper/1608.04914/) (2016)
- [Hilbert geometry of the Siegel disk: The Siegel-Klein disk model](https://mlsanity.com/paper/2004.08160/) (2020)
- [Geometric Matrix Midranges](https://mlsanity.com/paper/1907.04188/) (2019)
- [Inductive Geometric Matrix Midranges](https://mlsanity.com/paper/2006.01508/) (2020)
- [Online learning of Riemannian hidden Markov models in homogeneous Hadamard spaces](https://mlsanity.com/paper/2102.07771/) (2021)
- [Conal Distances Between Rational Spectral Densities](https://mlsanity.com/paper/1708.02818/) (2017)
- [Geometric Learning of Hidden Markov Models via a Method of Moments Algorithm](https://mlsanity.com/paper/2207.00818/) (2022)
- [Affine-Invariant Midrange Statistics](https://mlsanity.com/paper/2206.13766/) (2022)

## Built on by (most cited first, 5 of 5)

- [SPD Matrix Learning for Neuroimaging Analysis: Perspectives, Methods, and Challenges](https://mlsanity.com/paper/2504.18882/) (2025)
- [Approximation and bounding techniques for the Fisher-Rao distances between parametric statistical models](https://mlsanity.com/paper/2403.10089/) (2024)
- [Hilbert geometry of the symmetric positive-definite bicone: Application to the geometry of the extended Gaussian family](https://mlsanity.com/paper/2508.14369/) (2025)
- [Geometric statistics with subspace structure preservation for SPD matrices](https://mlsanity.com/paper/2407.03382/) (2024)
- [Geometric structures and deviations on James' symmetric positive-definite matrix bicone domain](https://mlsanity.com/paper/2603.02483/) (2026)

## Abstract (arXiv)

Differential geometric approaches to the analysis and processing of data in the form of symmetric positive definite (SPD) matrices have had notable successful applications to numerous fields including computer vision, medical imaging, and machine learning. The dominant geometric paradigm for such applications has consisted of a few Riemannian geometries associated with spectral computations that are costly at high scale and in high dimensions. We present a route to a scalable geometric framework for the analysis and processing of SPD-valued data based on the efficient computation of extreme generalized eigenvalues through the Hilbert and Thompson geometries of the semidefinite cone. We explore a particular geodesic space structure based on Thompson geometry in detail and establish several properties associated with this structure. Furthermore, we define a novel iterative mean of SPD matrices based on this geometry and prove its existence and uniqueness for a given finite collection of points. Finally, we state and prove a number of desirable properties that are satisfied by this mean.