Manifolds.jl: An Extensible Julia Framework for Data Analysis on Manifolds

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 B\'ezier 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.