Diffusion Approximations to Schrödinger Bridges on Manifolds
Garrett Mulcahy, Soumik Pal
TL;DR
The paper develops a diffusion-approximation framework for Schrödinger bridges on manifolds at small temperature, with strong results when the marginals coincide and the reference diffusion is reversible. It establishes a precise symmetric relative-entropy closeness between the Schrödinger bridge and a diffusion limit, derives a generator-transfer principle under curvature conditions, and proves next-order convergence of Schrödinger potentials, including manifold-specific curvature corrections in Hessian geometry. It then extends to different marginals via Brenier-type transforms, highlighting the role of the Hessian structure (and the Mirror Langevin diffusion) in approximating Euclidean Schrödinger bridges with quadratic costs. The work combines heat-kernel asymptotics, curvature-dimension theory, and optimal-transport tools to quantify the diffusion-regularized transport on manifolds and to connect these bridges to practical diffusion samplers and score-function limits.
Abstract
We present a collection of explicit diffusion approximations to small temperature Schrödinger bridges on manifolds. Our most precise results are when both marginals are the same and the Schrödinger bridge is on a manifold with a reference process given by a reversible diffusion. In the special case that the reference process is the manifold Brownian motion, we use the small time heat kernel asymptotics to show that the gradient of the corresponding Schrödinger potential converges in $L^2$, as the temperature vanishes, to a manifold analogue of the score function of the marginal. As an application of the previous result we show that the Euclidean Schrödinger bridge, computed for the quadratic cost, between two different marginal distributions can be approximated by a transformation of a two point distribution of a stationary Mirror Langevin diffusion.