# Geomstats¶

Geomstats is an open-source Python package for computations and statistics on nonlinear manifolds. The mathematical definition of manifold is beyond the scope of this documentation. However, in order to use Geomstats, you can visualize it as a smooth subset of the Euclidean space. Simple examples of manifolds include the sphere or the space of 3D rotations.

Data from many application fields are elements of manifolds. For instance, the manifold of 3D rotations SO(3), or the manifold of 3D rotations and translations SE(3), appear naturally when performing statistical learning on articulated objects like the human spine or robotics arms. Other examples of data that belong to manifolds are introduced in our paper.

Computations on manifolds require special tools of differential geometry. Computing the mean of two rotation matrices \(R_1, R_2\) as \(\frac{R_1 + R_2}{2}\) does not generally give a rotation matrix. Statistics for data on manifolds need to be extended to “geometric statistics” to perform consistent operations.

In this context, Geomstats provides code to fulfill four objectives:

provide educational support to learn “hands-on” differential geometry and geometric statistics, through its examples and visualizations.

foster research in differential geometry and geometric statistics by providing operations on manifolds to gain intuition on results of a research paper;

democratize the use of geometric statistics by implementing user-friendly geometric learning algorithms using Scikit-Learn API; and

provide a platform to share learning algorithms on manifolds.

The source code is freely available on GitHub.

The package is organized into two main modules: geometry and learning.

The module geometry implements concepts in differential geometry, such as manifolds and Riemannian metrics, with associated exponential and logarithmic maps, geodesics, and parallel transport.

The module learning implements statistics and learning algorithms for data on manifolds, such as estimation, clustering and dimension reduction. The code is object-oriented and classes inherit from scikit-learn’s base classes and mixins.

In both modules, the operations are vectorized for batch computation and provide support for different execution backends—namely NumPy, PyTorch, and TensorFlow.

To learn how to use geomstats, visit the page First steps. To contribute to geomstats visit the page Contributing.