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

What the math gives to ML

The paper offers a transferable mechanism for generating samples that remain exactly on a smooth embedded manifold: formulate the diffusion intrinsically using tangent-space noise and Riemannian drift, then learn the minimum-control stochastic bridge from a simple source distribution to an energy-weighted target. The important neural-network asset is not merely manifold-aware coordinates, but state-space feasibility enforced throughout the stochastic trajectory rather than repaired after sampling. This can become a constrained diffusion sampler for sphere-, Stiefel-, orthogonality-, or normalized-embedding-valued outputs, with a control network whose output is projected into the tangent space. The key falsifiable signatures are zero normal constraint violation, a control-energy/KL tradeoff predicted by Schrödinger bridge optimality, and improved sampling compared with unconstrained diffusion followed by projection.