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

TL;DR

The paper introduces Manifolds.jl and the accompanying ManifoldsBase.jl as a fast, extensible framework for data analysis on Riemannian manifolds and Lie groups in Julia. It presents a unified, decorator-enabled API that supports multiple metrics, representations, and in-place versus allocating computations, with practical demonstrations in Bézier splines, manifold optimization, and tangent-space PCA. Comprehensive benchmarks show competitive to superior performance versus Python/ Matlab libraries and robust accuracy across a range of manifolds, especially at low dimensions, while highlighting trade-offs at very high dimensions where TensorFlow-based backends may outperform. The work enables geometry-aware data analysis in a high-level, expressive language, facilitating integration with existing Julia ecosystems and promoting broader adoption of manifold methods.

Abstract

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.

Figures (6)

• Figure 1: Unit sphere in $\mathbb{R}^{3}$ with two points $p$, $q$. Tangent space at $p$ is labelled $T_{p}\mathbb{S}^{2}$, geodesic between $p$ and $q$ is the blue arc $\gamma(p, q)$, and the vector $X$ is the tangent vector from $T_{p}\mathbb{S}^{2}$ that points towards $q$. $Y$ is another vector from $T_{p}\mathbb{S}^{2}$ orthogonal to $X$. All tangent vectors are blue. Parallel transports of $X$ and $Y$ from $p$ to $q$ are shown as vector $\operatorname{PT}_{q \gets p} X$ and $\operatorname{PT}_{q \gets p} Y$, respectively.
• Figure 2: Illustration of the De Casteljau algorithm to evaluate a Bézier curve on the Sphere $\mathbb{S}^{2}$. The initial points and their shortest geodesics (blue) yield three points (cyan) and two points (green) during the recursion, and their connecting geodesic the point on the Bézier curve (orange).
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