Fast and Robust Shortest Paths on Manifolds Learned from Data
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Proceedings of Machine Learning Research, PMLR 89:15061515, 2019.
Abstract
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 wellbehaved Jacobians of the ODE, and usually, manifolds learned from data imply unstable and illconditioned Jacobians. Instead, we propose a fixedpoint 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 generalpurpose stateoftheart solvers as well as over specialized solvers.
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