Obtain a two-sided local remainder estimate for the Sinkhorn divergence

Establish a two-sided remainder estimate of order o(d_S^2(\mu,\nu)) for the local approximation of the Sinkhorn divergence S_\varepsilon(\mu,\nu) by the geodesic Sinkhorn distance d_S^2(\mu,\nu), rather than an upper bound with a remainder measured in the stronger one-Wasserstein distance W_c.

Background

The paper proves a lower asymptotic bound S_\varepsilon(\mu,\nu) \geq d_S2(\mu,\nu)+o(d_S2(\mu,\nu)) as d_S(\mu,\nu)\to0. For the corresponding upper bound, however, the available estimate has a remainder o(W_c2(\mu,\nu)), where W_c is the one-Wasserstein distance induced by the RKHS metric d_c. Since W_c is not uniformly controlled by d_S, this does not yield a two-sided o(d_S2) expansion.

The authors identify the obstruction in the off-diagonal operator composition H_{\mu_s,\mu_t}H_{\mu_t}{-1}, which does not simplify when s and t differ and is not currently understood in Banach spaces built directly on the common RKHS H_c. Resolving this issue would sharpen the local relationship between the Sinkhorn divergence and the geodesic Sinkhorn metric.

References

Next, we state a similar result for S_\varepsilon, however in this case we are unable to obtain a two-sided o(\dS2(\mu,\nu)) for the remainder.

— Approximation and computation of the geodesic Sinkhorn distance  (2610.02007 - Lavenant et al., 1 Oct 2026) in Section 4, immediately before Theorem 4.4 (the theorem numbered `theorem:estimates_with_Wc`); see also Remark 4.5, “Why not a better remainder?”