Computing geodesics for Riemannian manifolds is a difficult task that often relies on numerical approximations. However, these approximations tend to be either numerically unstable, have slow convergence, or scale poorly with manifold dimension and number of grid points. We introduce a new algorithm called GEORCE that computes geodesics in a local chart via a transformation into a discrete control problem. We show that GEORCE has global convergence and quadratic local convergence. In addition, we show that it extends to Finsler manifolds. For both Finslerian and Riemannian manifolds, we thoroughly benchmark GEORCE against several alternative optimization algorithms and show empirically that it has a much faster and more accurate performance for a variety of manifolds, including key manifolds from information theory and manifolds that are learned using generative models.

翻译：计算黎曼流形的测地线是一项困难的任务，通常依赖于数值近似方法。然而，这些近似方法往往存在数值不稳定、收敛速度慢，或随流形维度和网格点数量增加而扩展性差的问题。我们提出了一种名为GEORCE的新算法，该算法通过将局部坐标图下的测地线计算转化为离散控制问题来进行求解。我们证明了GEORCE具有全局收敛性和局部二次收敛性。此外，我们还展示了该算法可推广至芬斯勒流形。对于芬斯勒流形和黎曼流形，我们将GEORCE与多种替代优化算法进行了全面基准测试，并通过实验证明，对于包括信息论中的关键流形以及使用生成模型学习的流形在内的多种流形，GEORCE在计算速度和精度上均表现出显著优势。