标题： 从数据中学习的流形上的快速且鲁棒的最短路径 显示英文标题

标题： Fast and Robust Shortest Paths on Manifolds Learned from Data

摘要： 我们提出了一种快速、简单且鲁棒的算法，用于计算从数据中学习到的黎曼流形上的最短路径和距离。 这相当于在边界条件下求解常微分方程组（ODE）。 在这里，标准求解器表现不佳，因为它们需要ODE的良好雅可比矩阵，而通常从数据中学习到的流形会导致不稳定的和病态的雅可比矩阵。 相反，我们提出了一种用于求解ODE的定点迭代方案，该方案避免了雅可比矩阵。 这提高了求解器的稳定性，同时降低了计算成本。 在涉及黎曼度量学习和深度生成模型的实验中，我们展示了在速度和稳定性方面相对于通用最先进的求解器以及专用求解器的显著改进。 显示英文摘要

摘要： 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.