For data given on a nonlinear space, like angles, symmetric positive matrices, the sphere, or the hyperbolic space, there is often enough structure to form a Riemannian manifold. We present the Julia package Manifolds.jl, providing a fast and easy to use library of Riemannian manifolds and Lie groups. 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 B\'ezier splines, an optimization task on manifolds, and a principal component analysis on nonlinear data. In a benchmark, Manifolds.jl outperforms existing packages in Matlab or Python by several orders of magnitude and is about twice as fast as a comparable package implemented in C++.

中文翻译：

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Manifolds.jl：用于流形数据分析的可扩展 Julia 框架

对于非线性空间上给出的数据，如角度、对称正矩阵、球体或双曲空间，通常有足够的结构来形成黎曼流形。我们展示了 Julia 包 Manifolds.jl，它提供了一个快速且易于使用的黎曼流形和李群库。我们引入了一个通用接口，可在 ManifoldsBase.jl 中使用，通过它可以实现新的流形、应用程序和算法。我们使用 B\'ezier 样条、流形优化任务和非线性数据的主成分分析演示了 Manifolds.jl 的效用。在基准测试中，Manifolds.jl 的性能比 Matlab 或 Python 中的现有包高几个数量级，并且比用 C++ 实现的类似包快两倍。