We introduce Geomstats, an open-source Python toolbox for computations and statistics on nonlinear manifolds, such as hyperbolic spaces, spaces of symmetric positive definite matrices, Lie groups of transformations, and many more. We provide object-oriented and extensively unit-tested implementations. Among others, manifolds come equipped with families of Riemannian metrics, with associated exponential and logarithmic maps, geodesics and parallel transport. Statistics and learning algorithms provide methods for estimation, clustering and dimension reduction on manifolds. All associated operations are vectorized for batch computation and provide support for different execution backends, namely NumPy, PyTorch and TensorFlow, enabling GPU acceleration. This paper presents the package, compares it with related libraries and provides relevant code examples. We show that Geomstats provides reliable building blocks to foster research in differential geometry and statistics, and to democratize the use of Riemannian geometry in machine learning applications. The source code is freely available under the MIT license at \url{geomstats.ai}.

中文翻译：

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Geomstats：机器学习中黎曼几何的 Python 包

我们介绍了 Geomstats，这是一个开源 Python 工具箱，用于计算和统计非线性流形，例如双曲空间、对称正定矩阵的空间、李群变换等等。我们提供面向对象且经过广泛单元测试的实现。其中，流形配备了黎曼度量系列，以及相关的指数和对数图、测地线和平行传输。统计和学习算法提供了对流形进行估计、聚类和降维的方法。所有相关操作都被矢量化以进行批量计算，并支持不同的执行后端，即 NumPy、PyTorch 和 TensorFlow，从而实现 GPU 加速。本文介绍了包装，将其与相关库进行比较并提供相关代码示例。我们展示了 Geomstats 提供了可靠的构建块，以促进微分几何和统计学的研究，并使黎曼几何在机器学习应用中的使用民主化。源代码在 MIT 许可下可在 \url{geomstats.ai} 上免费获得。