Manopt.jl

Optimization Algorithm on Riemannian Manifolds.

For a function $f: ℳ → ℝ$ that maps from a Riemannian manifold ℳ to the real line, this package aims to solve

Find the minimizer p on ℳ, that is, the (or a) point where f attains its minimum.

Manopt.jl provides

• A framework to implement arbitrary optimization algorithms on Riemannian Manifolds
• A library of optimization algorithms on Riemannian manifolds
• an easy-to-use interface for (debug) output and recording values during an algorithm run.
• several tools to investigate the algorithms, gradients, and optimality criteria

Getting started

In Julia you can get started by just typing

using Pkg; Pkg.add("Manopt");

and then checkout the Get started: optimize! tutorial.

Related packages

Manopt.jl is based on ManifoldsBase.jl, hence the algorithms can be used with any manifold following this interface for defining a Riemannian manifold.

The following packages are related to Manopt.jl

• Manifolds.jl: a library of manifolds implemented using ManifoldsBase.jl :octocat: GitHub repository
• ManifoldsDiff.jl: a package to use (Euclidean) AD tools on manifolds, that also provides several differentials and gradients. :octocat: GitHub repository
• JuMP.jl: can be used as interface to solve an optimization problem with Manopt. See usage examples. :octocat: GitHub repository

Citation

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

@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 please cite for example

@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 to cite

@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 Julia Framework for Data Analysis on Manifolds},
    VOLUME    = {49},
    YEAR      = {2023}
}

as well. Note that all citations are in BibLaTeX format.

Further and Similar Packages & Links

Manopt.jl belongs to the Manopt family:

• www.manopt.org: the MATLAB version of Manopt, see also their :octocat: GitHub repository
• www.pymanopt.org: the Python version of Manopt—providing also several AD backends, see also their :octocat: GitHub repository

but there are also more packages providing tools on manifolds:

• Jax Geometry (Python/Jax): differential geometry and stochastic dynamics with deep learning
• Geomstats (Python with several backends): focusing on statistics and machine learning :octocat: GitHub repository
• Geoopt (Python & PyTorch): Riemannian ADAM & SGD. :octocat: GitHub repository
• McTorch (Python & PyToch): Riemannian SGD, Adagrad, ASA & CG.
• ROPTLIB (C++): a Riemannian OPTimization LIBrary :octocat: GitHub repository
• TF Riemopt (Python & TensorFlow): Riemannian optimization using TensorFlow

Did you use Manopt.jl somewhere? Let us know! We'd love to collect those here as well.