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We propose a fast, simple and robust algorithm for computing shortest pathsand distances on Riemannian manifolds learned from data. This amounts tosolving a system of ordinary differential equations (ODEs) subject to boundaryconditions. Here standard solvers perform poorly because they requirewell-behaved Jacobians of the ODE, and usually, manifolds learned from dataimply unstable and ill-conditioned Jacobians. Instead, we propose a fixed-pointiteration scheme for solving the ODE that avoids Jacobians. This enhances thestability of the solver, while reduces the computational cost. In experimentsinvolving both Riemannian metric learning and deep generative models wedemonstrate significant improvements in speed and stability over bothgeneral-purpose state-of-the-art solvers as well as over specialized solvers.

Fast and Robust Shortest Paths on Manifolds Learned from Data