# Manopt, a Matlab toolbox for optimization on manifolds

arXiv: https://arxiv.org/abs/1308.5200 · HTML: https://mlsanity.com/paper/1308.5200/

## Key facts

- Cited by 113 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Aug 2013.

## Built on by (most cited first, 30 of 113)

- [Edge Artificial Intelligence for 6G: Vision, Enabling Technologies, and Applications](https://mlsanity.com/paper/2111.12444/) (2021)
- [Universal Approximation Theorems for Differentiable Geometric Deep Learning](https://mlsanity.com/paper/2101.05390/) (2021)
- [Tangent Space Backpropagation for 3D Transformation Groups](https://mlsanity.com/paper/2103.12032/) (2021)
- [On Riemannian Optimization over Positive Definite Matrices with the Bures-Wasserstein Geometry](https://mlsanity.com/paper/2106.00286/) (2021)
- [Bayesian Learning for Neural Networks: an algorithmic survey](https://mlsanity.com/paper/2211.11865/) (2022)
- [Vector-valued Distance and Gyrocalculus on the Space of Symmetric Positive Definite Matrices](https://mlsanity.com/paper/2110.13475/) (2021)
- [Certifiably Optimal Outlier-Robust Geometric Perception: Semidefinite Relaxations and Scalable Global Optimization](https://mlsanity.com/paper/2109.03349/) (2021)
- [A Survey of Geometric Optimization for Deep Learning: From Euclidean Space to Riemannian Manifold](https://mlsanity.com/paper/2302.08210/) (2023)
- [PoLAR: Polar-Decomposed Low-Rank Adapter Representation](https://mlsanity.com/paper/2506.03133/) (2025)
- [Synchronization on circles and spheres with nonlinear interactions](https://mlsanity.com/paper/2405.18273/) (2024)
- [Understanding and Improving Shampoo and SOAP via Kullback-Leibler Minimization](https://mlsanity.com/paper/2509.03378/) (2025)
- [A Scalable Second Order Method for Ill-Conditioned Matrix Completion from Few Samples](https://mlsanity.com/paper/2106.02119/) (2021)
- [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)
- [Differentially private Riemannian optimization](https://mlsanity.com/paper/2205.09494/) (2022)
- [An Inexact Projected Gradient Method with Rounding and Lifting by Nonlinear Programming for Solving Rank-One Semidefinite Relaxation of Polynomial Optimization](https://mlsanity.com/paper/2105.14033/) (2021)
- [Learning with symmetric positive definite matrices via generalized Bures-Wasserstein geometry](https://mlsanity.com/paper/2110.10464/) (2021)
- [First-Order Algorithms for Min-Max Optimization in Geodesic Metric Spaces](https://mlsanity.com/paper/2206.02041/) (2022)
- [MMES: Mixture Model based Evolution Strategy for Large-Scale Optimization](https://mlsanity.com/paper/2203.12675/) (2022)
- [Chordal Averaging on Flag Manifolds and Its Applications](https://mlsanity.com/paper/2303.13501/) (2023)
- [Causal Abstraction Learning based on the Semantic Embedding Principle](https://mlsanity.com/paper/2502.00407/) (2025)
- [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)
- [Alignment of Density Maps in Wasserstein Distance](https://mlsanity.com/paper/2305.12310/) (2023)
- [Building Rome with Convex Optimization](https://mlsanity.com/paper/2502.04640/) (2025)
- [An Online Riemannian PCA for Stochastic Canonical Correlation Analysis](https://mlsanity.com/paper/2106.07479/) (2021)
- [Riemannian block SPD coupling manifold and its application to optimal transport](https://mlsanity.com/paper/2201.12933/) (2022)
- [Semantic keypoint-based pose estimation from single RGB frames](https://mlsanity.com/paper/2204.05864/) (2022)
- [NCVX: A General-Purpose Optimization Solver for Constrained Machine and Deep Learning](https://mlsanity.com/paper/2210.00973/) (2022)

## Abstract (arXiv)

Optimization on manifolds is a rapidly developing branch of nonlinear optimization. Its focus is on problems where the smooth geometry of the search space can be leveraged to design efficient numerical algorithms. In particular, optimization on manifolds is well-suited to deal with rank and orthogonality constraints. Such structured constraints appear pervasively in machine learning applications, including low-rank matrix completion, sensor network localization, camera network registration, independent component analysis, metric learning, dimensionality reduction and so on. The Manopt toolbox, available at www.manopt.org, is a user-friendly, documented piece of software dedicated to simplify experimenting with state of the art Riemannian optimization algorithms. We aim particularly at reaching practitioners outside our field.