Hard-Constrained Sampling on Embedded Riemannian Manifolds via Adjoint Schrödinger Bridges

Probability measures confined to nonlinear feasible sets arise whenever physical invariants or structural relations must hold exactly. Such laws are often known only up to an intractable normalizing constant, making sample generation challenging. Diffusion samplers offer an efficient alternative to MCMC methods, yet these formulations are developed primarily for Euclidean domains. Building upon the foundations of adjoint Schr\"odinger bridge sampling, this paper provides a theoretically justified method, through the lens of stochastic optimal control, to address this problem on smooth, compact, path-connected embedded Riemannian manifolds. As an element of novelty compared to existing literature, feasibility is enforced at the level of the state space, meaning the controlled diffusion is defined intrinsically on the curved space. Empirical validations are provided for several physics applications, including multimodal spherical distributions, Gibbs laws on Stiefel manifolds, closed-loop inverse kinematics, and robust Wahba optimization.