Fast and Robust Shortest Paths on Manifolds Learned from Data

TL;DR

This paper introduces a fast, stable, and efficient algorithm for computing shortest paths on data-learned Riemannian manifolds by avoiding Jacobian computations, outperforming existing solvers in speed and stability.

Contribution

It presents a novel fixed-point iteration method for solving ODEs on learned manifolds, improving robustness and computational efficiency over traditional solvers.

Findings

Significant speed improvements over state-of-the-art solvers

Enhanced stability in shortest path computations

Effective on both metric learning and deep generative models

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 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.