Manopt.jl

Optimization on Riemannian manifolds

Efficient algorithms for minimizing a function on a Riemannian manifold

Get started Available Solvers Developer Guide

Generic Implementations

Algorithms are implemented on a general Riemannian manifold using the interface from ManifoldsBase.jl.

🧩

Composable

All components are designed in a modular fashion, for example problems, objectives, and solver states, but also stopping criteria and step sizes. They can be reused and combined easily. Points and tangent vectors can be represented using arbitrary Julia types.

⚡️

Efficient

All methods use in-place evaluations for both the objective and the manifold. There is also an advanced, but easy to use caching system further promoting efficiency. The package is compatible with running on a GPU when you use ManifoldsGPU.jl.

📚

Well-documented and -tested

All algorithms are documented – both their theoretical foundation and all their options and keywords. The theoretical background also refers to further literature.

🛠️

Customizable

All solvers bring recording, debug and callback capabilities, the objectives can keep track of internal function calls.

Use with Manifolds.jl

If you have a certain manifold in mind, check out whether it is already provided in Manifolds.jl or LieGroups.jl.

For a function $f:\mathcal M → ℝ$ defined on a Riemannian manifold $\mathcal M$ algorithms in this package aim to solve

\[\operatorname*{argmin}_{p ∈ \mathcal M} f(p),\]

or in other words: find the point $p$ on the manifold $\mathcal M$, where $f$ reaches its minimal function value.

Manopt.jl provides a framework for optimization on manifolds as well as a library of optimization algorithms in Julia. It belongs to the Manopt family, which includes Manopt (Matlab) and pymanopt.org (Python), both aiming to provide the same framework in the flavour of the corresponding language.

Get Started

To get started with Manopt.jl, start Julia and type

] add Manopt

to install the package. Then you can dive directly into optimization on manifolds, following the 🏔️ Get started with Manopt.jl tutorial.

Manopt.jl makes it easy to use an algorithm for your favourite manifold as well as a manifold for your favourite algorithm. It already provides many manifolds and algorithms, which can easily be enhanced, for example to record certain data or debug output throughout iterations.

If you use Manopt.jl in your work, please cite the following:

Bergmann, R. (2022). Manopt.jl: Optimization on Manifolds in Julia, Journal of Open Source Software, 7(70), 3866.

doi: 10.21105/joss.03866

@article{Bergmann2022,
    Author    = {Ronny Bergmann},
    Doi       = {10.21105/joss.03866},
    Journal   = {Journal of Open Source Software},
    Number    = {70},
    Pages     = {3866},
    Publisher = {The Open Journal},
    Title     = {Manopt.jl: Optimization on Manifolds in {J}ulia},
    Volume    = {7},
    Year      = {2022},
}

To refer to a certain version or the source code in general, cite for example

Bergmann, R. (2026). Manopt.jl, Zenodo.

doi: 10.5281/zenodo.4290905

@software{manoptjl-zenodo-mostrecent,
    Author    = {Ronny Bergmann},
    Copyright = {MIT License},
    Doi       = {10.5281/zenodo.4290905},
    Publisher = {Zenodo},
    Title     = {Manopt.jl},
    Year      = {2024},
}

for the most recent version or a corresponding version specific DOI, see the list of all versions.

If you are also using Manifolds.jl, please consider citing

Axen, S. D., Baran, M., Bergmann, R., Rzecki, K. (2023). Manifolds.jl: An Extensible Julia Framework for Data Analysis on Manifolds, ACM Transactions on Mathematical Software, Volume 49, Issue 4, Article No. 33, pages 1–23.

doi: 10.1145/3618296, arXiv: 2106.08777

@article{AxenBaranBergmannRzecki:2023,
    AUTHOR    = {Axen, Seth D. and Baran, Mateusz and Bergmann, Ronny and Rzecki, Krzysztof},
    ARTICLENO = {33},
    DOI       = {10.1145/3618296},
    JOURNAL   = {ACM Transactions on Mathematical Software},
    MONTH     = {dec},
    NUMBER    = {4},
    TITLE     = {{Manifolds.jl}: An Extensible {J}ulia Framework for Data Analysis on Manifolds},
    VOLUME    = {49},
    YEAR      = {2023}
}

Main features

Optimization algorithms (solvers)

For every optimization algorithm, a solver is implemented based on an AbstractManoptProblem that describes the problem to solve and its AbstractManoptSolverState that sets up the solver, and stores values that are required between or for the next iteration.

Manifolds

This project is built upon ManifoldsBase.jl, a generic interface to implement manifolds. Certain functions are extended for specific manifolds from Manifolds.jl, but all other manifolds from that package can be used here, too.

The notation in the documentation aims to follow the notation of these packages.

Algorithm exploration

To visualize and interpret results, Manopt.jl provides a system to get debug output during the iterations of an algorithm as well as record capabilities, for example to record a specified tuple of values per iteration, most prominently RecordCost and RecordIterate. Take a look at the 🏔️ Get started with Manopt.jl tutorial on how to easily activate this.

Literature

If you want to get started with manifolds, a recommended reference is the book [Car92], and if you want to directly dive into optimization on manifolds, good references are [AMS08] and [Bou23], which are both available online for free.