GEORCE: a fast new control algorithm for computing geodesics.

Area of Science:

• Differential Geometry
• Numerical Analysis
• Optimization Theory

Background:

• Computing geodesics on Riemannian manifolds is challenging due to limitations of current numerical approximations.
• Existing methods often suffer from numerical instability, slow convergence, and poor scalability.
• There is a need for more robust and efficient algorithms for geodesic computation.

Purpose of the Study:

• Introduce a new algorithm, GEORCE, for computing geodesics on Riemannian and Finsler manifolds.
• Demonstrate the convergence properties and computational advantages of GEORCE.
• Benchmark GEORCE against existing algorithms across various manifold types.

Main Methods:

• GEORCE transforms the geodesic computation into a discrete control problem within a local chart.
• The algorithm is analyzed for its convergence properties, proving global and quadratic local convergence.
• Extensive benchmarking was performed on diverse manifolds, including those from information theory and generative models.

Main Results:

• GEORCE exhibits global convergence and quadratic local convergence, ensuring reliable computation.
• Empirical results show GEORCE significantly outperforms alternative optimization algorithms in speed and accuracy.
• The algorithm demonstrates effectiveness on both Riemannian and Finsler manifolds, including complex learned models.

Conclusions:

• GEORCE provides a stable, efficient, and accurate method for computing geodesics on Riemannian and Finsler manifolds.
• The algorithm's performance advantages make it suitable for complex applications in geometry and machine learning.
• GEORCE represents a significant advancement in the numerical computation of geodesics.