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.

我们推出 Julia 包 Manifolds.jl，提供快速且易于使用的黎曼流形和李群库。该软件包可以处理黎曼流形上定义的数据，例如圆、球、对称正定矩阵或双曲空间模型之一。我们引入了一个通用接口，可在 ManifoldsBase.jl 中使用，通过该接口可以实现新的流形、应用程序和算法。我们使用贝塞尔样条、流形优化任务以及非线性数据的主成分分析来演示 Manifolds.jl 的实用性。在基准测试中，Manifolds.jl 在速度方面优于所有类似的低维流形包；与 Python 和 Matlab 软件包相比，改进通常是几个数量级，而与 C/C++ 软件包相比，改进是两倍。对于高维流形，它的性能优于除 Tensorflow-Riemopt 之外的所有软件包，Tensorflow-Riemopt 是专门为高维流形定制的。