# Pymanopt: A Python Toolbox for Optimization on Manifolds using Automatic Differentiation

arXiv: https://arxiv.org/abs/1603.03236 · HTML: https://mlsanity.com/paper/1603.03236/

## Key facts

- Cited by 94 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Mar 2016.

## Built on by (most cited first, 30 of 94)

- [SPD domain-specific batch normalization to crack interpretable unsupervised domain adaptation in EEG](https://mlsanity.com/paper/2206.01323/) (2022)
- [Universal Approximation Theorems for Differentiable Geometric Deep Learning](https://mlsanity.com/paper/2101.05390/) (2021)
- [On Riemannian Optimization over Positive Definite Matrices with the Bures-Wasserstein Geometry](https://mlsanity.com/paper/2106.00286/) (2021)
- [Multiway Non-rigid Point Cloud Registration via Learned Functional Map Synchronization](https://mlsanity.com/paper/2111.12878/) (2021)
- [NRDF: Neural Riemannian Distance Fields for Learning Articulated Pose Priors](https://mlsanity.com/paper/2403.03122/) (2024)
- [HaLP: Hallucinating Latent Positives for Skeleton-based Self-Supervised Learning of Actions](https://mlsanity.com/paper/2304.00387/) (2023)
- [Geometry-aware Bayesian Optimization in Robotics using Riemannian Mat'ern Kernels](https://mlsanity.com/paper/2111.01460/) (2021)
- [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)
- [Symmetry Discovery Beyond Affine Transformations](https://mlsanity.com/paper/2406.03619/) (2024)
- [Nested Hyperbolic Spaces for Dimensionality Reduction and Hyperbolic NN Design](https://mlsanity.com/paper/2112.03402/) (2021)
- [Guiding Neural Collapse: Optimising Towards the Nearest Simplex Equiangular Tight Frame](https://mlsanity.com/paper/2411.01248/) (2024)
- [Federated Learning on Riemannian Manifolds](https://mlsanity.com/paper/2206.05668/) (2022)
- [Interactive Dimensionality Reduction for Comparative Analysis](https://mlsanity.com/paper/2106.15481/) (2021)
- [Riemannian Hamiltonian methods for min-max optimization on manifolds](https://mlsanity.com/paper/2204.11418/) (2022)
- [Horospherical Decision Boundaries for Large Margin Classification in Hyperbolic Space](https://mlsanity.com/paper/2302.06807/) (2023)
- [Learning with symmetric positive definite matrices via generalized Bures-Wasserstein geometry](https://mlsanity.com/paper/2110.10464/) (2021)
- [SECA: Semantically Equivalent and Coherent Attacks for Eliciting LLM Hallucinations](https://mlsanity.com/paper/2510.04398/) (2025)
- [Manifold optimization for non-linear optimal transport problems](https://mlsanity.com/paper/2103.00902/) (2021)
- [Feature Learning for Nonlinear Dimensionality Reduction toward Maximal Extraction of Hidden Patterns](https://mlsanity.com/paper/2206.13891/) (2022)
- [Wasserstein Gradient Flows for Optimizing Gaussian Mixture Policies](https://mlsanity.com/paper/2305.10411/) (2023)
- [Alignment of Density Maps in Wasserstein Distance](https://mlsanity.com/paper/2305.12310/) (2023)
- [Entangled Kernels -- Beyond Separability](https://mlsanity.com/paper/2101.05514/) (2021)
- [NCVX: A General-Purpose Optimization Solver for Constrained Machine and Deep Learning](https://mlsanity.com/paper/2210.00973/) (2022)
- [Medoid splits for efficient random forests in metric spaces](https://mlsanity.com/paper/2306.17031/) (2023)
- [Automatic differentiation for Riemannian optimization on low-rank matrix and tensor-train manifolds](https://mlsanity.com/paper/2103.14974/) (2021)
- [Optimization on manifolds: A symplectic approach](https://mlsanity.com/paper/2107.11231/) (2021)
- [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)
- [Geometric Neural Distance Fields for Learning Human Motion Priors](https://mlsanity.com/paper/2509.09667/) (2025)

## Abstract (arXiv)

Optimization on manifolds is a class of methods for optimization of an objective function, subject to constraints which are smooth, in the sense that the set of points which satisfy the constraints admits the structure of a differentiable manifold. While many optimization problems are of the described form, technicalities of differential geometry and the laborious calculation of derivatives pose a significant barrier for experimenting with these methods. We introduce Pymanopt (available at https://pymanopt.github.io), a toolbox for optimization on manifolds, implemented in Python, that---similarly to the Manopt Matlab toolbox---implements several manifold geometries and optimization algorithms. Moreover, we lower the barriers to users further by using automated differentiation for calculating derivative information, saving users time and saving them from potential calculation and implementation errors.