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arXiv Statistics ML By Tristan Luca Saidi, Florian Gunsilius, Gonzalo Mena

Second-order Sinkhorn Geometry and $\sqrt{n}$-Inference

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The paper develops a formal Riemannian framework for the Sinkhorn divergence, a debiased entropic optimal transport metric. It proves Hadamard differentiability of the Sinkhorn metric tensor, derives the Levi‑Civita covariant derivative, and obtains geodesic and parallel transport equations as ODEs on a fixed RKHS. Using Banach‑space ODE theory, the authors linearize these equations to establish √ n‑consistent Gaussian limits for plug‑in estimators of the exponential map, logarithmic map, parallel transport, and the induced Sinkhorn distance, and show bootstrap consistency for all these objects.

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