GEORCE: A Fast New Control Algorithm for Computing Geodesics

2025-05-09

AI Summary

Computing geodesics on Riemannian and Finsler manifolds typically relies on numerical approximations, which suffer from poor stability, slow convergence, and limited scalability to high dimensions. This paper introduces the first formulation of geodesic computation as a discrete optimal control problem, unified within a nonlinear optimization framework that incorporates manifold-aware gradient methods—applicable to both Riemannian and Finsler geometries. Theoretically, we establish global convergence and local quadratic convergence rates. Empirically, on high-dimensional manifolds arising in information geometry and generative modeling, our method achieves an average 2.3× speedup over state-of-the-art solvers, with superior accuracy, enhanced numerical stability, and milder growth in computational complexity with respect to dimensionality.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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 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.

Problem

• Computing geodesics for Riemannian manifolds efficiently
• Overcoming numerical instability and slow convergence issues
• Extending algorithm to Finsler manifolds for broader application

Innovation

• Transforms geodesic computation into control problem
• Ensures global and quadratic local convergence
• Extends to Finsler and Riemannian manifolds