Prove convergence of incremental Sinkhorn-chain refinement

Prove that incrementally refining the temporal discretization of the Sinkhorn chain and applying alternating minimization converges to high-quality approximations of geodesics for the geodesic Sinkhorn distance, with the refinement strategy converging faster than direct optimization at high temporal resolution.

Background

The numerical scheme begins with a coarse chain, optimizes its intermediate measures by alternating minimization, and then repeatedly doubles the number of time steps by inserting new measures. The authors motivate this procedure by the difficulty of propagating endpoint information through a poorly initialized fine chain.

The paper provides numerical and asymptotic justification for using vertical interpolation as an initialization at sufficiently fine resolution, but it does not establish convergence or a rate for the incremental refinement-and-optimization procedure itself.

References

We conjecture that incremental refinement of the chain will converge to high-quality candidates faster, since in a poorly initialized chain of high temporal resolution it will take a long time until information from the end points propagates through the whole chain by alternating optimization.

— Approximation and computation of the geodesic Sinkhorn distance  (2610.02007 - Lavenant et al., 1 Oct 2026) in Section 7.1, paragraph “Refining the chain,” immediately before Algorithm 1

However, we conjecture that in the continuous time limit of the restriction to measures with a fixed number of particles M, each particle will trace a clear path.

— Approximation and computation of the geodesic Sinkhorn distance  (2610.02007 - Lavenant et al., 1 Oct 2026) in Section 7.2, paragraph “Encoding and refining the chain”

However, it is unclear whether this is the minimum, or merely a saddle point of the energy functional \bar{E}N. Further research is required to settle this question.

— Approximation and computation of the geodesic Sinkhorn distance  (2610.02007 - Lavenant et al., 1 Oct 2026) in Section 7.3, paragraph “Geodesics between Gaussians”