Manifolds.jl: An Extensible Julia Framework for Data Analysis on Manifolds
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2021
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| _version_ | 1866910952007401472 |
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| author | Axen, Seth D. Baran, Mateusz Bergmann, Ronny Rzecki, Krzysztof |
| author_facet | Axen, Seth D. Baran, Mateusz Bergmann, Ronny Rzecki, Krzysztof |
| contents | We present the Julia package Manifolds$.$jl, providing a fast and easy-to-use library of Riemannian manifolds and Lie groups. This package enables working with data defined on a Riemannian manifold, such as the circle, the sphere, symmetric positive definite matrices, or one of the models for hyperbolic spaces. We introduce a common interface, available in ManifoldsBase$.$jl, with which new manifolds, applications, and algorithms can be implemented. We demonstrate the utility of Manifolds$.$jl using Bézier splines, an optimization task on manifolds, and principal component analysis on nonlinear data. In a benchmark, Manifolds$.$jl outperforms all comparable packages for low-dimensional manifolds in speed; over Python and Matlab packages, the improvement is often several orders of magnitude, while over C/C++ packages, the improvement is two-fold. For high-dimensional manifolds, it outperforms all packages except for Tensorflow-Riemopt, which is specifically tailored for high-dimensional manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2106_08777 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Manifolds.jl: An Extensible Julia Framework for Data Analysis on Manifolds Axen, Seth D. Baran, Mateusz Bergmann, Ronny Rzecki, Krzysztof Mathematical Software We present the Julia package Manifolds$.$jl, providing a fast and easy-to-use library of Riemannian manifolds and Lie groups. This package enables working with data defined on a Riemannian manifold, such as the circle, the sphere, symmetric positive definite matrices, or one of the models for hyperbolic spaces. We introduce a common interface, available in ManifoldsBase$.$jl, with which new manifolds, applications, and algorithms can be implemented. We demonstrate the utility of Manifolds$.$jl using Bézier splines, an optimization task on manifolds, and principal component analysis on nonlinear data. In a benchmark, Manifolds$.$jl outperforms all comparable packages for low-dimensional manifolds in speed; over Python and Matlab packages, the improvement is often several orders of magnitude, while over C/C++ packages, the improvement is two-fold. For high-dimensional manifolds, it outperforms all packages except for Tensorflow-Riemopt, which is specifically tailored for high-dimensional manifolds. |
| title | Manifolds.jl: An Extensible Julia Framework for Data Analysis on Manifolds |
| topic | Mathematical Software |
| url | https://arxiv.org/abs/2106.08777 |