Short and Straight: Geodesics on Differentiable Manifolds
This work addresses a foundational issue in machine learning for tasks such as reduced-order modeling and statistical inference, though it appears incremental as it builds on existing differential geometry tools.
The authors tackled the problem of computing valid geodesics and geodesic distances on differentiable manifolds discovered by machine learning models, proposing a method that leverages Hamiltonian-conserving geodesics and a curvature-based training mechanism, resulting in principled means for applications like latent-space interpolation and optimal control.
Manifolds discovered by machine learning models provide a compact representation of the underlying data. Geodesics on these manifolds define locally length-minimising curves and provide a notion of distance, which are key for reduced-order modelling, statistical inference, and interpolation. In this work, we first analyse existing methods for computing length-minimising geodesics. We find that these are not suitable for obtaining valid paths, and thus, geodesic distances. We remedy these shortcomings by leveraging numerical tools from differential geometry, which provide the means to obtain Hamiltonian-conserving geodesics. Second, we propose a model-based parameterisation for distance fields and geodesic flows on continuous manifolds. Our approach exploits a manifold-aware extension to the Eikonal equation, eliminating the need for approximations or discretisation. Finally, we develop a curvature-based training mechanism, sampling and scaling points in regions of the manifold exhibiting larger values of the Ricci scalar. This sampling and scaling approach ensures that we capture regions of the manifold subject to higher degrees of geodesic deviation. Our proposed methods provide principled means to compute valid geodesics and geodesic distances on manifolds. This work opens opportunities for latent-space interpolation, optimal control, and distance computation on differentiable manifolds.