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

Tristan Luca Saidi; Florian Gunsilius; Gonzalo Mena

stat.MLOct 27, 2026

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

Authors: Tristan Luca Saidi, Florian Gunsilius, Gonzalo Mena

Organizations: Department of Statistics and Data Science, Carnegie Mellon University · Department of Economics, Emory University

Abstract

The Sinkhorn divergence, a debiased form of entropic optimal transport, has been widely used in various scientific disciplines due to its computational and statistical tractability. Unlike the Wasserstein distance, however, the Sinkhorn divergence is not a metric and it does not naturally admit a formal Riemannian structure. These properties are central to many applications of optimal transport methods that utilize interpolation, barycentric averaging, parallel transport or other geometric operations. Recent work addresses these shortcomings by constructing a formal Riemannian tangent space and a metric tensor on the space of probability measures that quadratically approximates the Sinkhorn divergence. In this work, we extend this framework with two major threads of contributions. For a suitable class of finite-smoothness cost functions, we prove Hadamard differentiability of the Sinkhorn metric tensor and we derive the Levi-Civita covariant derivative in this geometry; we then use the covariant derivative to obtain the geodesic and parallel transport equations, each of which amount to an ODE on a fixed Reproducing Kernel Hilbert Space (RKHS) with smooth coefficients. Our second contribution is to linearize the solutions to the geodesic and parallel transport equations using standard tools from Banach space ODE theory. We show that under standard regularity conditions these linearizations imply $\sqrt{n}$ -limiting Gaussian processes centered at population values for the plugin estimators of the exponential map, the logarithmic map, parallel transport and the induced Sinkhorn distance. Moreover, we show that the nonparametric bootstrap is consistent for all objects mentioned. To the best of our knowledge, these are the first $\sqrt{n}$ --limit theorems and bootstrap consistency results for canonical geometric operations in a transport-based geometry on the space of probability measures.

Figures & tables

Figure 1: Eulerian chain approximation of a Sinkhorn geodesic on $X=[0,1]$ , computed using the numerical scheme of Lavenant et al. (2026b) . The endpoint densities are $\rho_{0}(x)=0.2+0.8\eta(x-0.25)$ and $\rho_{1}(x)=0.2+0.8\eta(x-0.75)$ , where $\eta$ is a normalized smooth bump supported on $[-0.1,0.1]$ . The approximation uses a Matérn- $5/2$ kernel, $\varepsilon=0.02$ , $500$ spatial cells, and $128$ temporal steps. Left: densities at selected interpolation times, with the dotted line indicating zero. Right: minimum density over the spatial grid as a function of time. We note that the intermediate densities remain strictly positive throughout the discretized interpolation.

	Sinkhorn	Wasserstein
Geodesics	$\displaystyle\begin{gathered}\Pi_{\mu_{t}}\Bigl(\ddot{\beta}_{t}+\tfrac{1}{2}S_{\mu_{t}}^{-1}\dot{S}_{\mu_{t}}\dot{\beta}_{t}+\Omega_{\dot{\beta}_{t}}\dot{\beta}_{t}\Bigr)=0\end{gathered}$ Second-order ODE in $\mathcal{H}_{c}$ .	$\displaystyle\begin{aligned} \partial_{t}\mu_{t}+\nabla\!\cdot(\mu_{t}v_{t})&=0,\\ \partial_{t}v_{t}+\nabla_{v_{t}}v_{t}&=0\end{aligned}$ First-order PDE system.
Parallel transport	$\displaystyle\begin{gathered}\Pi_{\mu_{t}}\Bigl(\dot{u}_{t}+\tfrac{1}{2}S_{\mu_{t}}^{-1}\dot{S}_{\mu_{t}}u_{t}+\Omega_{\dot{\beta}_{t}}u_{t}\Bigr)=0\end{gathered}$ First-order linear ODE in $\mathcal{H}_{c}$ .	$\displaystyle\begin{gathered}P_{\mu_{t}}\Bigl(\partial_{t}w_{t}+\nabla_{v_{t}}w_{t}\Bigr)=0\end{gathered}$ First-order PDE.

Table 1: Comparison of geodesic and parallel transport equations in Sinkhorn and Wasserstein geometry. For the Wasserstein equations listed, see Gigli (2012) or Ambrosio and Gigli (2008) .

Panel of Figure 2 — Figure 2: Visualization of the Sinkhorn exponential map in the $\mathcal{H}_{c}$ parameterization.

Panel of Figure 3 — Figure 3: Visualization of the coordinate chart $\psi:V_{R}\subset\mathfrak{T}_{\mu}\mathcal{P}(X)\to\mathbb{S}(\mathcal{H}_{c})$ .

Panel of Figure 4 — Figure 4: Notation for the exponential-map linearization on the chart $I_{t_{i}}$ .

Panel of Figure 5 — Figure 5: Visualization of the parallel imputation map, $\operatorname{Imp}(\mu,\nu,\rho).$

Panel of Figure 6 — Figure 6: Notation for the parallel transport / $\operatorname{Imp}$ linearization on the chart $I_{t_{i}}$ , in direct analogy with that of Figure 4 .

Optimal Transport

Optimal transport compares and transforms probability distributions by finding low-cost ways to move mass between them.

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Aug 13, 2026

Sinkhorn Linearization and the Spectral Proxy: Unifying the Statistical and Algorithmic Theory of Feature-Parameterized Inverse Optimal Transport via a Single Spectral Sandwich

We develop the statistical and algorithmic theory of inverse optimal transport (IOT) under the feature-parameterized cost C_theta(i,j) = -theta^T phi(i,j). The core technical contribution is the Sinkhorn linearization -- the implicit-function sensitivity of the entropic OT plan to the cost -- together with its spectral proxy, a formula that is spectrally exact yet geometrically transparent. The restricted Hessian on the tangent space satisfies the spectral sandwich (pi_min/epsilon) I <= H_T^{-1} <= (pi_max/epsilon) I, yielding the single core bound sigma_min >= (pi_min/(a_max epsilon)) sqrt(lambda_min(Sigma)) that drives the entire theory. On this core we establish four theorems and one observation. T1 (identifiability): theta is globally injective on the quotient of the gauge kernel, with dimension bound F <= (K-1)^2. T2 (sparsistency): the l1-penalized estimator recovers the true support under irrepresentability and score concentration, with exponential failure probability. T3 (well-posedness): the feature-moment map M(theta) = Phi^T x_theta is strongly monotone, and the inverse is Lipschitz with constant L <= epsilon ||Phi^T S_a||_op / (pi_min lambda_min(Sigma)). T4 (convergence): local strong convexity with mu >= pi_min^2 lambda_min(Sigma) / epsilon^2 guarantees monotone gradient descent convergence. O5 (misspecification): the estimator converges to the OT-model projection of the truth; the Holder continuity of the projection map is assessed numerically, yielding setting-dependent empirical exponents alpha_eff in (0,1).

Han Dong, Jiaming Li, Yongqiang Gong +2

School of Medicine, Nankai University · College of Artificial Intelligence, Nankai University

Inverse Optimal Transport Parameter Identifiability Optimal Transport

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Aug 29, 2026

Uniform Statistical Convergence of Empirical Sinkhorn Potentials with Exponential and Polynomial Dependence on the Regularization Parameter

We study the empirical Sinkhorn estimator of the entropic optimal transport potentials under the uniform loss. Since the potentials are only unique up to additive constants, we measure the error using the quotient supremum norm, defined as

d_\infty([u],[v]) = \inf_{a\in\mathbb{R}}\|u-v-a\|_\infty

. For a fixed regularization parameter

\varepsilon>0

, we establish a non-asymptotic statistical rate of

n^{-1/2}

. This is achieved by combining the Birkhoff-Hopf contraction theorem with entropy bounds on normalized kernel sections. However, the constant in this bound grows exponentially with

1/ε

. To improve this, we isolate geometric conditions under which the empirical estimator maintains the

n^{-1/2}

rate but features polynomial dependence on

1/\varepsilon

. The key requirement is a polynomial residual-stability estimate for the population Sinkhorn map. We provide sufficient criteria for this, including a polynomial contraction property and a local inverse estimate. Furthermore, we introduce two rigorously verifiable model classes an

\varepsilon

-weak residual-interaction class obtained after separable centering and another based on connected tight-edge graphs for fixed discrete costs where the polynomial rate is guaranteed without relying on abstract resolvent assumptions. Finally, we establish matching minimax lower bounds demonstrating that the

\varepsilon n^{-1/2}

rate cannot be uniformly improved in the bounded-interaction regime.

Denis Belomestny

Department of Mathematics, University of Duisburg-Essen

Optimal Transport Entropy Regularization Minimax Estimation

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Sep 22, 2026

Statistical Rates for Entropic Optimal Transport in the Discrete to SubGaussian Regime

We study statistical rates in entropic optimal transport in the semi-discrete regime where one measure has finite support and the other is subGaussian. Our main result establishes parametric convergence rates for the empirical dual potentials to their population counterparts, with no dimension dependence in the leading term. Our result relies on tailored strong concavity analysis of the semi-dual objective, coupled with specialized bounds for the semi-discrete potentials. As a consequence, we obtain fast rates for downstream quantities derived from the optimal coupling. Chiefly, the empirical barycentric projection achieves a squared-error rate

n^{-1}

, matching the fully compact case and improving over the less favorable

n^{-1/2}

rate known for fully subGaussian settings. Altogether, these results may indicate a lower complexity adaptation phenomenon whereby the statistical complexity of the barycentric projection is governed by the discrete measure. As an application, we analyze Sinkhorn-EM, an EM-type algorithm in which the E-step is replaced by an entropic optimal transport problem. In a well-specified and balanced two-component Gaussian mixture model, we prove

\sqrt{n}

-consistency of the empirical iterates to their population counterparts for any fixed number of iterations, matching classical EM rates up to a

\sqrt{\log n}

factor. Simulations support the theory.

Tomas Gonzalez, Gonzalo Mena

Department of Machine Learning, Carnegie Mellon University · Department of Statistics & Data Science, Carnegie Mellon University

Expectation-Maximization Optimal Transport

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Optimal Transport

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