# Levi-Civita Covariant Derivative

The Sinkhorn divergence of entropic optimal transport is fast and statistically well behaved, but it is not a metric and has no Riemannian structure; the earlier tangent space and metric tensor came without covariant derivative, geodesic or parallel transport equations, and inference on them was open. Under finite-smoothness costs the metric tensor is shown Hadamard differentiable along curves of measures, which yields the Levi-Civita covariant derivative and turns geodesics and parallel transport into ODEs with smooth coefficients on a fixed RKHS. Linearizing those solutions with Banach-space ODE theory gives √n Gaussian limits for plug-in estimators of the exponential map, logarithmic map, parallel transport and squared Sinkhorn distance, with a consistent nonparametric bootstrap.

The Sinkhorn divergence of entropic optimal transport is fast and statistically well behaved, but it is not a metric and has no Riemannian structure; the earlier tangent space and metric tensor came without covariant derivative, geodesic or parallel transport equations, and inference on them was open. Under finite-smoothness costs the metric tensor is shown Hadamard differentiable along curves of measures, which yields the Levi-Civita covariant derivative and turns geodesics and parallel transport into ODEs with smooth coefficients on a fixed RKHS. Linearizing those solutions with Banach-space ODE theory gives √n Gaussian limits for plug-in estimators of the exponential map, logarithmic map, parallel transport and squared Sinkhorn distance, with a consistent nonparametric bootstrap.