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

Tristan Luca Saidi, Florian Gunsilius, Gonzalo Mena

arXiv:2610.36488v1stat.MLcs.LGmath.STstat.TH

TL;DR

The Sinkhorn divergence lacks the metric and Riemannian structure needed for geometric operations on probability measures. This paper extends its geometry and establishes Gaussian limits and bootstrap consistency for plug-in estimators of canonical operations.

  • Problem

    The Sinkhorn divergence does not provide a metric or Riemannian structure, while statistical estimation for associated geometric functionals can be computationally and statistically prohibitive.

    Source: §1 · pp. 1–2

  • Method

    For finite-smoothness costs, the authors prove Hadamard differentiability of the metric tensor, derive the Levi-Civita covariant derivative, and linearize geodesic and parallel transport equations.

    Source: §1 · p. 2

  • Results

    Under regularity conditions, plug-in estimators of the exponential map, logarithmic map, parallel transport, and Sinkhorn distance have root-n Gaussian limits, and the nonparametric bootstrap is consistent.

    Source: §1 · p. 2

  • Takeaways & Limitations

    These results provide a tractable statistical foundation for uncertainty quantification of geometric estimands using Sinkhorn geometry as a surrogate for Wasserstein geometry.

    Source: §1 · p. 2

    A lower bound on density removes many geometric singularities and boundary issues; whether this assumption can be weakened remains open.

    Source: §4 · p. 41

Abstract

from arXiv · show

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.

1 Introduction

This section introduces second-order geometry for the Sinkhorn divergence and statistical inference for its canonical geometric operations, including Gaussian limits and bootstrap consistency for plug-in estimators. It also specifies finite-smoothness RKHS conditions supporting the covariant derivative and notes a limitation of the squared Euclidean cost.

1 Introduction

  • Hadamard differentiability of the Sinkhorn metric tensor enables derivation of the Levi-Civita covariant derivative, geodesic equations, and parallel transport equations under finite-smoothness cost conditions.The exponential map and parallel transport each reduce to an ODE on a fixed RKHS with smooth coefficients.Source: §1 · p. 2
  • The plug-in exponential map, logarithmic map, parallel transport, and Sinkhorn distance have √n-Gaussian limits, and the nonparametric bootstrap consistently approximates their limiting distributions.The results establish the first √n-limit theorems and bootstrap consistency results for canonical geometric operations in a transport-based geometry on probability measures.Source: §1 · pp. 2–3

1.1 Related Work

  • Lavenant et al. (2026a) construct the Sinkhorn Riemannian metric and geodesic distance; this work derives its covariant derivative, geodesic and parallel transport equations, and exponential and logarithmic maps.Prior work includes discretization schemes for the geodesic Sinkhorn distance (Lavenant et al., 2026b) and inference for EOT quantities, which are distinct from these Riemannian operations.Source: §1.1 · p. 4

1.3.1 The Sinkhorn Divergence

  • The Sinkhorn divergence vanishes only when its measures agree, but its square root fails the triangle inequality; its favorable √n-rate estimability motivates a local Riemannian approximation.The approximation uses a metric tensor and tangent space based on the second-order behavior of the divergence.Source: §1.3.1 · pp. 6, 8

1.4 Preliminaries on Reproducing Kernel Hilbert Spaces (RKHS)

  • Under Assumption 1.7, spectral decay yields an RKHS Hc norm-equivalent to Hs(X), with sufficient regularity for the cost and kernel to belong to C(m,m)(X × X).The condition is s > m + d/2, where m governs the perturbation and statistical theory.Source: §1.4 · p. 9
  • Because Hc is a Banach algebra under pointwise multiplication, the Levi–Civita covariant derivative is a bounded operator on the tangent space.The multiplication property follows from Hc being norm-equivalent to Hs(X) with s > d/2.Source: §1.4 · p. 10
  • Although the tangent space starts from m-th order distributional perturbations, its metric completion is governed by the Sobolev regularity of the transport-kernel RKHS.Source: §1.4 · p. 10
  • The squared Euclidean cost is inadmissible under these finite-smoothness assumptions because its Gaussian-kernel RKHS is not closed under pointwise multiplication.In contrast, under Assumption 1.7, Hc is norm-equivalent to a Sobolev space with s > d/2 and is a Banach algebra.Source: §1.4 · p. 10
  • Matérn kernels satisfying ϑ > m and ϑ + d/2 ∈ N meet Assumption 1.7 and induce costs that approximate the squared Euclidean cost as ϑ → ∞.Source: §1.4 · p. 10

2 Sinkhorn Riemannian Geometry

This section develops the Sinkhorn metric’s operator representation and differentiability to establish regularity tools for constructing a metric-compatible, torsion-free Levi-Civita covariant derivative. It uses this derivative to characterize geodesics and parallel transport through local ODE formulations in a fixed RKHS, with stated regularity conditions and limitations.

2.3 Deriving the Covariant Derivative

  • The covariant-derivative construction seeks a metric-compatible, torsion-free derivative under sufficient regularity; the stated assumptions are sufficient conditions, not claimed necessities.The uniform positive-density lower bound is the strongest restriction; singular or vanishing-density measures may require additional analysis because their admissible perturbations can depend on the measure.Source: §2 · p. 11, §2.3 · p. 15

2.1 Parameterizing the Action of ˜gµ

  • The metric tensor is represented by a bounded self-adjoint operator Sµ on the fixed RKHS Hc, with ˜gµ(v, w) = ε 2⟨Sµv, w⟩Hc.The restriction of Sµ to TµP(X) is also self-adjoint.Source: §2.1 · pp. 11–12

2.2 Operator Differentiability

  • Differentiating the metric along a curve of measures reduces to differentiating Sµ on the fixed RKHS Hc, yielding a time derivative by the Hadamard chain rule.The paper identifies Hadamard differentiability as a shared framework for geometric and statistical results, although suitable differentiability along curves alone would suffice for this time derivative.Source: §2.2 · p. 12, §2.2.1 · p. 14

2.2.1 Hadamard Differentiability

  • The map µ ↦ Sµ is Hadamard differentiable tangentially to Tmµ, and this also makes the metric tensor Hadamard differentiable.This differentiability is established in a normed measure space and is intended to support functional-delta-method inference for plug-in estimators.Source: §2.2 · p. 12, §2.2.1 · p. 14
  • For H1 curves µt with positive density, the metric tensor is time differentiable almost everywhere, providing regularity needed to verify metric compatibility.The result also gives time differentiability of ˜gµt(vt, wt) for H1 tangent curves vt and wt.Source: §2.2.1 · p. 14

2.3.1 Regular Curves

  • First-order Sinkhorn regular curves are H1 in Hm(X)*, while second-order curves additionally require their Hc-level velocity to be a Sinkhorn regular tangent field.The first-order condition is stronger than the regularity implied by requiring Bµ to be H1 in Hc.Source: §2.3.1 · p. 16

2.3.2 The Covariant Derivative

  • Finite-smoothness costs ensure the RKHS can represent the second-order objects needed for the classical derivative; infinitely smooth costs may fail this requirement.The induced RKHS must be closed under pointwise multiplication for the correction functional to be bounded and admit a Riesz representative; the Matérn cost has this property and can approximate squared Euclidean cost arbitrarily well.Source: §2.3.2 · pp. 17–18
  • Under Assumption 2.5, the constructed covariant derivative is metric-compatible and torsion-free, defining the Levi-Civita derivative for the Sinkhorn metric.It is identified within a family of metric-compatible derivatives by a correction term that enforces torsion-freeness.Source: §2.3.2 · pp. 17–18

2.4 The Geodesic Equations and Parallel Transport

  • Geodesics are characterized by vanishing covariant acceleration, while parallel tangent fields are characterized by vanishing covariant derivative along the curve.The paper specializes these characterizations to derive the Sinkhorn geodesic and parallel transport equations.Source: §2.4 · pp. 19–20
  • The geodesic ODE does not itself ensure nonnegative measures or action minimization, and its solutions need not remain in the probability-measure space.The equations give a local extension of the infinitesimal geometry to the ambient RKHS; variational minimizers satisfying the stated assumptions provide probability-measure-valued geodesics.Source: §2.4 · pp. 19, 21
  • A variational minimizer with uniformly positive density that is second-order Sinkhorn-regular satisfies the geodesic equations and is a Sinkhorn geodesic.The density lower bound is imposed on the minimizing interpolant; it is not known to follow from the endpoint conditions.Source: §2.4 · pp. 19–20
  • Under Assumption 2.5, geodesic initial-value problems and parallel transport have unique local solutions; transport requires a second-order Sinkhorn-regular base curve.Unlike the Wasserstein setting described here, the fixed-RKHS formulation needs no analogous spatial Lipschitz condition on the tangent velocity; temporal regularity suffices for the Banach-space ODE.Source: §2.4 · p. 21

3 Limit Theorems via Linearization

This section develops differentiability and linearization results for Sinkhorn geometric operations to establish plug-in Gaussian limit theory. Under stated regularity conditions, it also establishes bootstrap consistency for geodesics, logarithmic maps, Sinkhorn distance, and parallel imputation, and describes extensions to distribution-valued curves and functionals.

3 Limit Theorems via Linearization

  • Linearizing the geometric operations and applying the functional delta method yields √n-Gaussian limits for plug-in geodesic interpolants, the logarithmic map, Sinkhorn Riemannian distance, and parallel imputation.The geodesic and parallel-transport solution maps are treated through smooth extensions and Banach-space ODE theory; the variationally defined distance receives a separate Hadamard differentiability argument.Source: §3 · p. 22

3.1 The Exponential Map

  • On its stated domain, the entire geodesic trajectory map is Hadamard differentiable, with derivative given by a constrained linear variational initial-value problem.The domain requires a tangent initial velocity, an existing geodesic on [0, 1], and a trajectory contained in U+; the variation is tangent to the sphere along the path and uses initial conditions h0 = Dγ Bµ and ḣ0 = a.Source: §3.1 · p. 24

3.2 The Logarithmic Map

  • On the regular domain, the Sinkhorn logarithmic map is Hadamard differentiable, with regularity requiring a unique logarithmic velocity and an invertible endpoint derivative.The result applies to a variational minimizing geodesic satisfying Proposition 2.12’s assumptions; the derivative is obtained by differentiating the exponential-map endpoint relation and accounting for variation of the tangent space.Source: §3.2 · pp. 28–29
  • The logarithmic map is Hadamard differentiable at a nonconjugate pair (µ, ν), tangentially to T_m.The implicit-function argument gives a locally unique C1 logarithmic map near the reference velocity; it does not rule out other initial velocities outside that neighborhood.Source: §3.2 · p. 30

3.3 Parallel Transport

  • The parallel-imputation map Imp is Hadamard differentiable at (µ, ν, ρ), tangentially to Tm, when (µ, ρ) and (µ, ν) are nonconjugate in Dlog, (ν, w) lies in Dexp, and the associated geodesics satisfy Proposition 2.12’s assumptions.The limit theorem applies even when the population imputation trajectory leaves B[P(X)].Source: §3.3 · pp. 31–32

3.4 Sinkhorn Distance

  • The squared Sinkhorn distance is fully Hadamard differentiable with a bounded linear derivative under the stated smoothness, nonconjugacy, and minimizing-geodesic conditions.The first-variation argument uses admissible interpolant competitors and strong stability to avoid differentiating the minimizing curves; endpoint perturbations determine the variation.Source: §3.4 · p. 35

3.5 Central Limit Theorems

  • Joint empirical-process convergence and bounded linear Hadamard derivatives imply plug-in limit distributions by the functional delta method.The three independent empirical processes converge to independent tight Brownian bridges when m > d/2.Source: §3.5 · p. 35

3.5.1 Geodesics, logarithmic map and Sinkhorn distance

  • The geodesic results assume a uniformly positive Lebesgue density along the variational geodesic, second-order Sinkhorn regularity, and (GE).The passage defines v = logµ(ν) as the initial velocity of that geodesic.Source: §3.5.1 · p. 36
  • For m > d/2 and the stated nonconjugacy and regularity assumptions, the geodesic, logarithmic map, and Sinkhorn-distance estimators have joint centered Gaussian limits.The joint limit includes the geodesic path, logarithmic map, and distance-related components.Source: §3.5.1 · pp. 36–37
  • The two plug-in estimators of squared Sinkhorn distance differ by oP(n^-1/2) and share the same limiting Gaussian distribution, despite potentially differing at finite sample sizes.One evaluates the empirical metric tensor at the empirical logarithmic map; the other evaluates the intrinsic Sinkhorn distance between empirical measures.Source: §3.5.1 · p. 37
  • Distribution-coordinate geodesic limits hold in a Sobolev dual space, but the empirical geodesic need not be a finite signed measure; even the population imputation curve need not be one.Smooth functionals of these curves still admit delta-method limits in the stated Sobolev-dual setting.Source: §3.5.1 · p. 37, §3.5.2 · p. 39

Then the bootstrap functional delta method gives

  • The nonparametric bootstrap consistently estimates the joint limit for the geodesic, logarithmic map, and Sinkhorn distance, and extends to smooth functionals of the geodesic.The functional extension applies under the same assumptions as the geodesic limit theorem.Source: §3.5.1 · p. 38, Then the bootstrap functional delta… · p. 38

3.5.2 Parallel imputation via parallel transport

  • Under m > d/2 and the assumptions of Theorem 3.4 at (µ, ν, ρ), the parallel-imputation limit is a centered Gaussian random element in C1([0, 1]; Hc).Source: §3.5.2 · p. 39
  • The nonparametric bootstrap is consistent for parallel imputation and extends to its induced distribution-valued curve and fully Hadamard differentiable functionals.Bootstrap inference for these functionals does not require the imputed curves to be probability-measure-valued.Source: §3.5.2 · p. 40

4 Discussion

The discussion identifies computational approximations and the density lower bound as open directions for the Sinkhorn geometry. The appendices define differentiability notions and verify a symmetry property of the Schrödinger potentials.

4 Discussion

  • Finite-dimensional approximations with error oP(n^-1/2) could transfer the established limiting distributions and bootstrap guarantees to computational implementations.The authors also identify uncertainty quantification for distributional interpolation and trajectory estimation as a potential application.Source: §4 · pp. 40–41
  • Whether the density lower bound is intrinsic remains open, particularly near measures whose densities vanish.The bound removes many singularities and boundary issues in the geometry.Source: §4 · p. 41

A.1 Normed-space Differentiability

  • The appendix distinguishes Gâteaux differentiability from Hadamard differentiability and states the functional delta method for Hadamard directionally differentiable maps.The delta method transfers weak convergence of scaled inputs to weak convergence after applying the map.Source: §A.1 · p. 42, Hadamard differentiability · p. 42

Hadamard differentiability.

  • Hadamard differentiability requires the directional derivative to be linear, with convergence tested along perturbation sequences whose directions converge.The appendix contrasts this stronger notion with directional differentiability.Source: Hadamard differentiability · p. 42

B.1 Proof of Lemma 1.1

  • Symmetry of the cost and uniqueness of normalized Schrödinger potential pairs imply f_µ,ν = g_ν,µ.The proof first relates the swapped potentials up to constants, then shows the constants vanish.Source: §B.1 · p. 43

B.2 Proofs for Section 2.1

The appendix establishes that the operators used in the Sinkhorn geometry are well defined and that the metric tensor has the stated operator properties. It also proves invertibility of S_µ and self-adjointness on the tangent space.

B.2 Proofs for Section 2.1

  • The operators R_µ and J_µ are independent of the chosen representative from Q_µ.For J_µ, adding a constant to a representative does not change its value.Source: §B.2 · p. 43
  • The map Θ_µ from H_µ modulo constants to the tangent space is well defined and admits an inverse.The proof shows the construction does not depend on the representative of the equivalence class.Source: §B.2 · pp. 43–44
  • The operator S_µ is invertible on Hc, with its inverse established through operator identities involving J_µ, Q_µ, and A_µ.The proof verifies both left- and right-inverse properties.Source: §B.2 · pp. 44–45

B.2.1 Proof of Theorem 2.1

  • The metric tensor operator S_µ is self-adjoint on the tangent space T_µP(X).The proof first shows S_µ maps the tangent space onto itself and then uses symmetry of the metric tensor.Source: §B.2.1 · p. 46

B.2.2 Supplemental Results

This supplemental section establishes operator identities, regularity results, and proof steps underlying the paper’s geometry, including results on tangent-space characterization and differentiability of the potential map and metric tensors. It also establishes that ∇Ω is a valid covariant derivative along a curve when µ̇_t = 0 and that Πµ is the ˜gµ-orthogonal projection onto the tangent space.

B.2.2 Supplemental Results

  • Sµ maps the tangent space onto itself, while Sµ, Sµ^-1, and (along admissible paths) its time derivative are self-adjoint on Hc.The supplemental results also identify the pairing involving JµRµAµ with Rµ on L2(µ)/R and establish an intertwining relation for jµ and the resolvent operators.Source: §B.2.2 · pp. 47–48
  • For c ∈ C(m,m)(X × X), the Schrödinger-potential map is Hadamard differentiable, and the change-of-variable map is Hadamard differentiable as well.The finite-smoothness extension weakens the smooth-cost result to order-m regularity; the change-of-variable result follows from the potential-map differentiability.Source: §B.2.2 · pp. 48, 49, 51

B.3 Proof of Proposition 2.2

  • The proof characterizes the statistical tangent space as Hm(X)*_0 and establishes a continuous dense embedding of the Riemannian tangent space into Hs(X)*_0.The argument uses the positive-density assumption and Sobolev-space embeddings.Source: §B.3 · p. 51

B.4 Proof of Proposition 2.4

  • The self-Schrödinger potential map is Hadamard differentiable by composing the two-measure potential map with the diagonal map µ ↦ (µ, µ).The diagonal map has derivative γ ↦ (γ, γ), and the chain rule applies with Proposition B.9.Source: §B.4 · pp. 51–52

B.5 Proof of Theorem 2.2

  • Hadamard differentiability of Jµ and Rµ implies differentiability of Sµ, and composing Sµ with the metric-tensor pairing yields differentiability of the metric tensor.The composition S(A, B) = idHc + 2AB has derivative 2 ȦB + 2AḂ.Source: §B.5 · p. 52

B.6 Proof of Theorem 2.3

  • For almost every t, µt is differentiable in Cm(X)* with derivative in the statistical tangent space, yielding time differentiability of Sµt by the chain rule.The proof uses µt(1X) = 1, the positive-density assumption, and the differentiability result for the metric operator.Source: §B.6 · p. 52
  • For H1 paths vt and wt, the metric tensor along (µt, vt, wt) has a Hadamard derivative obtained by applying the chain rule to the metric-tensor differentiability result.The path derivatives use the weak time derivatives of vt and wt.Source: §B.6 · pp. 52–53

B.7 Proof of Theorem 2.4

  • When µ̇_t = 0, ∇Ω is a valid covariant derivative along the curve; the proof verifies linearity over R and the Leibniz rule.It does not verify the third property referenced in Lee (2018, Theorem 4.24).Source: §B.7 · p. 53

B.7.1 Supplemental Results

  • Πµ is the ˜gµ-orthogonal projection from Hc onto TµP(X).The proof uses that Πµ is the Hc-orthogonal projection and shows it is self-adjoint with respect to ˜gµ.Source: §B.7.1 · p. 54

B.8 Proof of Theorem 2.5

The proof establishes metric compatibility and torsion-freeness for the proposed covariant derivative. It extends torsion-freeness from smooth bounded zero-mass density perturbations to all tangent directions.

B.8 Proof of Theorem 2.5

  • The proposed covariant derivative is metric compatible and torsion-free, and therefore is the Levi-Civita derivative associated with the Sinkhorn metric.Torsion-freeness is first shown for smooth bounded zero-mass density perturbations, then extended by continuity and density to all tangent directions under Assumption 2.5.Source: §B.8 · pp. 55–56

B.8.1 Supplemental Results

The supplemental results establish symmetry of the metric-tensor derivative term and characterize its tangent-space behavior, while the proof of Lemma 2.7 establishes that the derivative of B_mu is a bounded linear bijection on the tangent space.

B.8.1 Supplemental Results

  • C_mu(u, v)(w) is symmetric in u and v because the Hadamard derivative of S_mu is self-adjoint.The identity is expressed as C_mu(u, v)(w) = ⟨(D_sigma_w S_mu)u, v⟩ = ⟨u, (D_sigma_w S_mu)v⟩.Source: §B.8.1 · p. 56
  • Subtracting the two identities shows that the difference of the initial derivatives is orthogonal to B_mu and therefore belongs to T_mu P(X).The conclusion uses symmetry of the H_c inner product.Source: §B.8.1 · p. 57

B.9 Proof of Lemma 2.7

  • The derivative D(·)B_mu is a bounded linear bijection from T_mu P(X) onto itself.The proof then identifies its inverse.Source: §B.9 · p. 57

B.10 Proof of Proposition 2.13

The proof establishes local existence and uniqueness for both the geodesic equation and parallel transport by expressing them as Banach-space ODEs with suitable regularity.

B.10 Proof of Proposition 2.13

  • The geodesic equation is a first-order ODE with a C1 vector field in tangent-bundle coordinates, yielding a unique local solution for each coordinate initial condition.The initial condition (0, v) corresponds to the ambient initial state (B_mu, v).Source: §B.10 · p. 57
  • In a local chart, parallel transport becomes an ODE with smooth coefficients, bounded linear in the transported vector, and has a unique local solution.The solution pulls back to the unique local parallel tangent field with the prescribed initial value.Source: §B.10 · p. 58

B.11 Additional Results for Section 2.3

These additional results connect regular measure-valued curves to admissible RKHS curves and provide further operator and metric-tensor identities.

B.11 Additional Results for Section 2.3

  • Under Assumption 2.5, every first-order Sinkhorn regular curve is admissible; in particular, an H1 measure path induces an H1 curve in H_c.The proof uses Lipschitz continuity of B and the composition property for absolutely continuous curves.Source: §B.11 · p. 58
  • The kernel relation k_mu(x, y) = a_mu(x)a_mu(y)k_c(x, y) yields H_mu(sigma) = a_mu H_c(a_mu sigma).Source: §B.11 · p. 58
  • For tangent vectors u and v, C_mu(u, v) has a Riesz representative in the tangent space.The displayed claim identifies this representative, and the proof notes that its first argument is tangent.Source: §B.11 · pp. 58, 60

B.12 Additional Results for Section 2.4

A sufficiently smooth curve in the stronger dual Sobolev space has a continuous derivative in the weaker dual space, and hence belongs to its H1 path space.

B.12 Additional Results for Section 2.4

  • If d/2 < r < m < s, p(m − r) > s − m, and mu is Cp+1 as an Hs(X)*-valued curve, then mu is C1 and H1 as an Hm(X)*-valued curve.The conclusion transfers regularity from the stronger dual Sobolev space to the weaker one.Source: §B.12 · p. 60
  • Sobolev interpolation gives continuity of t ↦ mu_t in Hm(X)* from continuity in Hs(X)*.Source: §B.12 · p. 60
  • Uniformly convergent finite-difference approximations establish continuity of the derivative in Hm(X)* on the full interval.The proof uses forward differences on the first half and backward differences on the second half.Source: §B.12 · p. 61

B.12.1 Proof of Proposition 2.12 (a)

The proof establishes sufficient regularity for a minimizing path and then shows that the path satisfies the geodesic equations. The argument uses a variational criterion and smoothness of the fixed-RKHS parameterization.

B.12.1 Proof of Proposition 2.12 (a)

  • Admissible perturbations of the minimizer yield a variational criterion by differentiating the pulled-back energy.The perturbations preserve probability measures for sufficiently small perturbation size, and minimality makes the first variation vanish.Source: §B.12.1 · p. 62, §B.12.2 · p. 64
  • The variational identity is represented by a bounded linear functional and converted into a strong-form equation using smoothness of the operator field.Uniform coercivity permits inversion along the path; the resulting regularity upgrades the path to W 2,∞ and then to smoothness.Source: §B.12.1 · pp. 62–63
  • The minimizing path is second-order Sinkhorn regular.The proof first establishes first-order regularity, then shows the RKHS-valued curve and its velocity are smooth enough for second-order regularity.Source: §B.12.1 · p. 63

B.12.2 Proof of Proposition 2.12 (b)

  • Differentiating the variational criterion and using self-adjointness and commuting mixed derivatives cancels terms to recover the geodesic equation.The identity holds against every element of the tangent space, yielding the geodesic equations in (GE).Source: §B.12.2 · pp. 64–65

B.13.1 Main results

This section assembles Hadamard differentiability results for measure-dependent components used in the Sinkhorn geometry. The results cover scaling functions, projection and operator maps, including the metric-related operators.

B.13.1 Main results

  • The scaling function a_mu and its pointwise reciprocal e_mu are Hadamard differentiable as maps into C^m(X).The proof obtains the reciprocal result by composition with smooth exponential maps and the Hadamard chain rule.Source: §B.13.1 · pp. 65–66
  • The map from the measure to e_mu is also Hadamard differentiable with values in the fixed RKHS Hc.The proof uses a direct analytical argument for this RKHS-valued result.Source: §B.13.1 · p. 66
  • The projection operator Pi_mu is Hadamard differentiable, and its derivative combines with that of J_mu to give differentiability of K_mu.K_mu is represented as Pi_mu J_mu, so the product rule supplies its derivative.Source: §B.13.1 · pp. 67–68
  • The metric-related operators J_mu, K_mu, and R_mu are Hadamard differentiable with respect to the measure.Their derivatives follow by combining differentiability of constituent maps with bounded operator multiplication and inversion.Source: §B.13.1 · pp. 68–69
  • The inverse-operator construction for R_mu uses invertibility of id_Hc − K_mu and the differentiability rule for inversion.The proof also uses R_mu = R_mu Pi_mu to obtain the derivative expression.Source: §B.13.1 · p. 69

B.13.2 Supplemental results

Supplemental results establish operator identities and inverse bounds used in the Sinkhorn geometry. The geodesic equations are also expressed as a state-space ODE in RKHS variables.

B.13.2 Supplemental results

  • The operators id_Qmu + K_mu and id_Qmu − K_mu are invertible on H_mu/R, and the inverse defining Q_mu has norm at most 1.The spectral bound for K_mu on the quotient space is the stated basis for invertibility.Source: §B.13.2 · p. 70
  • The operator R_mu is the inverse of id_Hc − K_mu and satisfies R_mu = R_mu Pi_mu.The proof establishes both right- and left-inverse properties.Source: §B.13.2 · pp. 70–71
  • The operator K_mu on Hc factors as Pi_mu J_mu.A related quotient-space representation is K_mu = L_mu J_mu.Source: §B.13.2 · p. 71

C.1 Proof of Lemma 3.2

  • The geodesic equations are characterized by a state-space ODE in the variables (p_t, u_t).These variables are formed by projecting the RKHS curve and its velocity onto the measure-dependent tangent space.Source: §C.1 · p. 73

C.2 Proof of the Stitching Lemma

This section proves the regularity and differentiability properties needed for the geometric results, including smoothness and well-posedness of chart equations and stitched geodesic flows. It also establishes convergence of variational minimizers, Hadamard differentiability of the logarithmic-map norm, and chartwise variation relations.

C.2 Proof of the Stitching Lemma

  • Finite chart-flow stitching makes the geodesic trajectory map C1 and Hadamard differentiable, yielding the exponential-map derivative with h0 = DγBµ and hdot0 = a.The construction composes finitely many smooth local flows and chart transitions along the population trajectory.Source: §C.2 · p. 74

C.3 Extension and Smoothness of ODE Coefficients

  • Smooth constructions in β-coordinates transfer through the local inverse change of variables to distribution-indexed operators, and establish smooth geodesic and parallel-transport ODE coefficients.The constructions use the domain and kernel conditions in Assumption 2.5.Source: §C.3 · p. 74

C.3.1 The Ambient Domain U+

  • The ambient sets V+ and U+ are open, B[P(X)] lies in U+, and the relevant maps are smooth, with Jβ self-adjoint and equal to Kβ.For probability measures, minX Bµ ≥ kmin/√kmax and ∥JBµ∥op < 1.Source: §C.3.1 · p. 75

C.3.2 Inverting the change of variables

  • M is a smooth local diffeomorphism on U+ with a smooth inverse, and its extended operators are smooth with derivatives agreeing at µ with their previously derived Hadamard derivatives.When Assumption 2.5.3 holds, the restriction is the unique continuous extension of the statistical Hadamard derivative.Source: §C.3.2 · pp. 76, 78

C.3.3 Smoothness of Coefficients

  • The geodesic coefficients are smooth and Γβ is bounded bilinear with values in β⊥; the extended connection is metric-compatible, and ambient geodesics have constant metric speed.Metric compatibility holds along every C1 curve in U+ ∩ S(Hc).Source: §C.3.3 · pp. 79–80
  • The chart acceleration is smooth and quadratic in velocity, yielding unique local ambient geodesics and solutions on [0, 1] for sufficiently small initial velocities; the constrained linearized geodesic equation is unique.The chart parallel-transport field is also smooth and bounded linear in its transported vector argument.Source: §C.3.3 · pp. 81–82

C.4 Proof of Lemma 3.7

  • The proof obtains the normal component of the transported-vector derivative by differentiating the tangency condition, then converts the result to chart state-space coordinates.Source: §C.4 · p. 82

C.5 Proof of Lemma 3.8

  • Chartwise first variations are propagated using spatial derivatives of the state equations and chart-transition differentials, with initial perturbations converted from Hc coordinates.Source: §C.5 · pp. 82–83

C.6 Proof of Lemma 3.9

  • Variational minimizers converge strongly in H1([0, 1]; Hs(X)*), and the reweighting perturbation has first-order direction σt = (ut − μt(ut))μt.Source: §C.6 · pp. 83–84

C.7 Proof of Lemma 3.10

  • Since δ > 0 is arbitrary, the result establishes full Hadamard differentiability.Source: §C.7 · p. 85

C.8 Additional Results for Section 3.5

The supplementary results establish that the induced Sinkhorn distance agrees to second order with the metric tensor, and that geodesics have constant metric speed. They also give Gaussian-process and stability results for the statistical tangent space and variational geodesics.

C.8 Additional Results for Section 3.5

  • d2 S(µ, ν) equals ˜gµ(v, v), and the metric speed of a geodesic remains constant along its trajectory.The speed is identified with its initial value when v = ˙β0.Source: §C.8 · p. 86
  • For m > d/2, the µ-Brownian bridge has a tight, Borel measurable realization in the statistical tangent space and takes values almost surely in a separable subset.Under the canonical embedding into ℓ∞(Bm), its restriction to Bm is a centered tight Gaussian process with covariance Covµ(f, g).Source: §C.8 · pp. 86–87
  • When the variational minimizer joining µ and ν is unique, minimizers between weakly converging endpoint measures converge uniformly in dS to it.The limiting metric geodesic is shown to be a variational minimizer before uniqueness is applied.Source: §C.8 · pp. 87–88
  • Under the same uniqueness assumptions, the corresponding transformed curves βn converge strongly to β in H1([0, 1]; Hc).Here βn_t and βt are the transforms of the approximating and limiting minimizing curves.Source: §C.8 · p. 88

C.9 Supplemental Results

The supplemental results establish smoothness of the operators used in the geometric construction and show that the relevant charts are diffeomorphisms. They also establish bounded linearity of the derivatives for the exponential map and parallel transport imputation.

C.9 Supplemental Results

  • Jβ, Kβ, Rβ, and Cβ(u, v)♯ depend smoothly on their arguments in the stated operator and function spaces.In particular, Kβ = Jβ on V+; Rβ and Cβ(u, v)♯ are smooth on their stated domains.Source: §C.9 · p. 89
  • The sphere chart ψ and induced tangent-bundle chart Ψ are diffeomorphisms onto their images, with bounded linear isomorphisms as derivatives.The inverse of ψ on its image is given by ψ−1(β) = Πµβ.Source: §C.9 · p. 90
  • The exponential map is fully Hadamard differentiable, with a bounded linear derivative in its joint perturbation.The trajectory variation is bounded and linear into C1([0, 1]; Hc).Source: §C.9 · p. 91
  • The parallel imputation variation is bounded and linear in the perturbations of its inputs.The variation of the parallel field and its endpoint value are bounded linear maps.Source: §C.9 · p. 91

Appendix D Miscellaneous Results

Appendix D establishes Sobolev embeddings and nonlinear-operation regularity used in the paper’s analysis. The Proposition 1.6 proof applies kernel smoothness and RKHS inclusions to the relevant transport kernels.

Appendix D Miscellaneous Results

  • The Sobolev embeddings include dense Cm(X) into Hm(X), an injection Hm(X)* into Cm(X)*, and a dense continuous Hs(X) into Cm(X) when s > m + d/2.Duality also gives a continuous injection Cm(X)* into Hs(X)*.Source: Appendix D · p. 92
  • Pointwise and dual multiplication are bounded bilinear maps on Hc, while pointwise exponentiation is C∞ with kth derivative ε^-k exp(f/ε) times the product of its directions.The same assertions hold on Hs(X) and its dual.Source: Appendix D · pp. 92–93
  • If u belongs to Hs(X), s > d/2, and u ≥ δ > 0, then both 1/u and log u belong to Hs(X).The proof uses the continuous embedding into C(X) and the Sobolev composition theorem.Source: Appendix D · p. 93

D.1 Proof of Proposition 1.6

  • For Proposition 1.6, finite smoothness of c yields corresponding smoothness of k_c, and the RKHS inclusions place Hc and Hμ continuously in Cm(X).The proof also uses smoothness of the self-transport potential to establish smoothness of the associated kernel.Source: §D.1 · p. 94
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