# geomstats: a Python Package for Riemannian Geometry in Machine Learning

arXiv: https://arxiv.org/abs/1805.08308 · HTML: https://mlsanity.com/paper/1805.08308/

## Key facts

- Cited by 71 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 May 2018.

## Built on by (most cited first, 30 of 71)

- [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022)
- [Generalized Shape Metrics on Neural Representations](https://mlsanity.com/paper/2110.14739/) (2021)
- [Universal Approximation Theorems for Differentiable Geometric Deep Learning](https://mlsanity.com/paper/2101.05390/) (2021)
- [Universal Approximation Under Constraints is Possible with Transformers](https://mlsanity.com/paper/2110.03303/) (2021)
- [GeomNet: A Neural Network Based on Riemannian Geometries of SPD Matrix Space and Cholesky Space for 3D Skeleton-Based Interaction Recognition](https://mlsanity.com/paper/2111.13089/) (2021)
- [Geometric Autoencoders -- What You See is What You Decode](https://mlsanity.com/paper/2306.17638/) (2023)
- [A Survey of Geometric Optimization for Deep Learning: From Euclidean Space to Riemannian Manifold](https://mlsanity.com/paper/2302.08210/) (2023)
- [Beyond Euclid: An Illustrated Guide to Modern Machine Learning with Geometric, Topological, and Algebraic Structures](https://mlsanity.com/paper/2407.09468/) (2024)
- [Hyperbolic Deep Learning for Foundation Models: A Survey](https://mlsanity.com/paper/2507.17787/) (2025)
- [Symmetry Discovery Beyond Affine Transformations](https://mlsanity.com/paper/2406.03619/) (2024)
- [Small Transformers Compute Universal Metric Embeddings](https://mlsanity.com/paper/2209.06788/) (2022)
- [HyperCore: The Core Framework for Building Hyperbolic Foundation Models with Comprehensive Modules](https://mlsanity.com/paper/2504.08912/) (2025)
- [Geometric Neural Diffusion Processes](https://mlsanity.com/paper/2307.05431/) (2023)
- [SECA: Semantically Equivalent and Coherent Attacks for Eliciting LLM Hallucinations](https://mlsanity.com/paper/2510.04398/) (2025)
- [Differentially Private Fr'echet Mean on the Manifold of Symmetric Positive Definite (SPD) Matrices with log-Euclidean Metric](https://mlsanity.com/paper/2208.04245/) (2022)
- [Closed-form geodesics and trust-region method to calculate Riemannian logarithms on Stiefel and its quotient manifolds](https://mlsanity.com/paper/2103.13327/) (2021)
- [NCVX: A General-Purpose Optimization Solver for Constrained Machine and Deep Learning](https://mlsanity.com/paper/2210.00973/) (2022)
- [Fast hyperboloid decision tree algorithms](https://mlsanity.com/paper/2310.13841/) (2023)
- [Learning Distributions on Manifolds with Free-Form Flows](https://mlsanity.com/paper/2312.09852/) (2023)
- [HypLL: The Hyperbolic Learning Library](https://mlsanity.com/paper/2306.06154/) (2023)
- [Polynomial Chaos Expansions on Principal Geodesic Grassmannian Submanifolds for Surrogate Modeling and Uncertainty Quantification](https://mlsanity.com/paper/2401.16683/) (2024)
- [Riemann2: Learning Riemannian Submanifolds from Riemannian Data](https://mlsanity.com/paper/2503.05540/) (2025)
- [SPD Matrix Learning for Neuroimaging Analysis: Perspectives, Methods, and Challenges](https://mlsanity.com/paper/2504.18882/) (2025)
- [tensorflow-riemopt: A Library for Optimization on Riemannian Manifolds](https://mlsanity.com/paper/2105.13921/) (2021)
- [NCVX: A User-Friendly and Scalable Package for Nonconvex Optimization in Machine Learning](https://mlsanity.com/paper/2111.13984/) (2021)
- [Neural Networks as Paths through the Space of Representations](https://mlsanity.com/paper/2206.10999/) (2022)
- [Differential geometry with extreme eigenvalues in the positive semidefinite cone](https://mlsanity.com/paper/2304.07347/) (2023)
- [BlackDAN: A Black-Box Multi-Objective Approach for Effective and Contextual Jailbreaking of Large Language Models](https://mlsanity.com/paper/2410.09804/) (2024)
- [Stiefel Flow Matching for Moment-Constrained Structure Elucidation](https://mlsanity.com/paper/2412.12540/) (2024)
- [Riemannian MeanFlow for One-Step Generation on Manifolds](https://mlsanity.com/paper/2603.10718/) (2026)

## Abstract (arXiv)

We introduce geomstats, a python package that performs computations on manifolds such as hyperspheres, hyperbolic spaces, spaces of symmetric positive definite matrices and Lie groups of transformations. We provide efficient and extensively unit-tested implementations of these manifolds, together with useful Riemannian metrics and associated Exponential and Logarithm maps. The corresponding geodesic distances provide a range of intuitive choices of Machine Learning loss functions. We also give the corresponding Riemannian gradients. The operations implemented in geomstats are available with different computing backends such as numpy, tensorflow and keras. We have enabled GPU implementation and integrated geomstats manifold computations into keras deep learning framework. This paper also presents a review of manifolds in machine learning and an overview of the geomstats package with examples demonstrating its use for efficient and user-friendly Riemannian geometry.