Fast and Robust Shortest Paths on Manifolds Learned from Data
This addresses a computational bottleneck for researchers and practitioners in machine learning working with manifold-based methods, though it appears incremental as it builds on existing ODE solving techniques.
The paper tackles the problem of computing shortest paths on data-learned Riemannian manifolds, where standard ODE solvers fail due to unstable Jacobians, and proposes a fixed-point iteration scheme that avoids Jacobians, resulting in significant improvements in speed and stability over state-of-the-art solvers.
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.