We propose a fast, simple and robust algorithm for computing shortest paths and distances on Riemannian manifolds learned from data. This amounts to solving a system of ordinary differential equations (ODEs) subject to boundary conditions. Here standard solvers perform poorly because they require well-behaved Jacobians of the ODE, and usually, manifolds learned from data imply unstable and ill-conditioned Jacobians. Instead, we propose a fixed-point iteration scheme for solving the ODE that avoids Jacobians. This enhances the stability of the solver, while reduces the computational cost. In experiments involving both Riemannian metric learning and deep generative models we demonstrate significant improvements in speed and stability over both general-purpose state-of-the-art solvers as well as over specialized solvers.

我们提出了一种快速、简单、健壮的算法，用于计算从数据中学习的黎曼流形上的最短路径和距离。该算法利用常微分方程组（ODE）求解，通过固定点迭代方案来提高求解器的稳定性和降低计算成本。在黎曼度量学习和深度生成模型的实验中，我们证明了该算法相比于最先进的通用求解器和专用求解器在速度和稳定性方面的重大改进。

从数据学习流形快速可靠的最短路径