What Makes a Good Path? Factoring Manifold Support and Path Geometry

The manifold hypothesis suggests that real-world data concentrates on low-dimensional structures, yet navigating meaningful trajectories on latent manifolds remains an open challenge. A common approach is to construct density-aware metrics, but such constructions bundle two distinct questions: (i) where valid data lies and (ii) how we choose to traverse on the manifold. Since different tasks may demand different traversal behaviors even on the same data manifold, specifying these two aspects independently offers additional modeling flexibility. We propose to factor the path objective into a kinetic term governed by a freely chosen geometric prior and a score-based potential, derived from the negative log-density of a pretrained diffusion model, that acts as a soft constraint for manifold adherence. The potential encourages dynamics to evolve along the data support, while the geometric prior remains free to encode task-specific notions of path optimality. To solve the resulting optimization problem, we introduce \textit{geodesic force matching} (GFM), a direct-collocation algorithm that discretizes the trajectory and optimizes all waypoints jointly to satisfy the Euler--Lagrange force balance in a least-squares sense. On synthetic manifolds and 3D shape interpolation, our method produces smooth, on-manifold paths and performs favorably against recent density-aware baselines, most notably in the low-noise regime where those baselines become numerically unstable.