GEORCE: A Fast New Control Algorithm for Computing Geodesics

TL;DR

GEORCE is a novel algorithm that efficiently computes geodesics on Riemannian and Finsler manifolds, offering global and quadratic local convergence, outperforming existing methods in speed and accuracy across various applications.

Contribution

The paper introduces GEORCE, a new control-based algorithm for geodesic computation with proven convergence properties and broad applicability to different manifold types.

Findings

GEORCE achieves faster convergence than existing algorithms.

GEORCE provides more accurate geodesic computations.

Benchmark results show superior performance on diverse manifolds.

Abstract

Computing geodesics for Riemannian manifolds is a difficult task that often relies on numerical approximations. However, these approximations tend to be either numerically unstable, have slow convergence, or scale poorly with manifold dimension and number of grid points. We introduce a new algorithm called GEORCE that computes geodesics in a local chart via a transformation into a discrete control problem. We show that GEORCE has global convergence and quadratic local convergence. In addition, we show that it extends to Finsler manifolds. For both Finslerian and Riemannian manifolds, we thoroughly benchmark GEORCE against several alternative optimization algorithms and show empirically that it has a much faster and more accurate performance for a variety of manifolds, including key manifolds from information theory and manifolds that are learned using generative models.