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arxiv: 2604.12940 · v1 · submitted 2026-04-14 · 🧮 math.ST · stat.TH

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Distributional Convergence of Empirical Entropic Optimal Transport and Statistical Applications

Santiago Arenas-Velilla , Axel Munk , Luis-Alberto Rodr\'iguez

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Pith reviewed 2026-05-10 13:56 UTC · model grok-4.3

classification 🧮 math.ST stat.TH
keywords entropic optimal transportempirical processesHadamard differentiabilitydelta methodcolocalization analysisweak convergencebootstrap consistencystatistical inference
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The pith

Empirical entropic optimal transport plans satisfy asymptotic weak convergence for a broad class of functionals including colocalization processes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that functionals of the empirical EOT plan converge weakly to a Gaussian limit under regularity conditions on the cost, entropy parameter, and marginal distributions. This is proved by verifying Hadamard differentiability of the functional at the population level and invoking the extended delta method on the empirical measures. The result directly supports statistical procedures such as uniform confidence bands for colocalization curves and consistent bootstrap inference. A reader would care because EOT plans already appear in applications like quantifying protein proximity in cell biology, yet lacked a distributional theory for uncertainty quantification until now. The work therefore turns a computationally convenient transport tool into one that can support formal inference.

Core claim

We derive asymptotic weak convergence results for a large class of functionals of the EOT plan, in which the colocalization process is included. The proof is based on Hadamard differentiability and the extended delta method.

What carries the argument

Hadamard differentiability of the map from marginal measures to the functional of the EOT plan, followed by the extended delta method applied to the empirical marginals.

If this is right

  • Uniform confidence bands can be constructed for colocalization curves derived from the EOT plan.
  • Bootstrap consistency holds for the class of Hadamard differentiable functionals of the empirical EOT plan.
  • The same limiting distribution applies to any functional satisfying the differentiability condition, not only the colocalization process.
  • The theory covers both the optimal transport plan itself and derived summary statistics under the given regularity on cost and entropy.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same differentiability-plus-delta-method route could be checked for other regularized transport problems such as those using different divergences.
  • Viewing the EOT plan as a random measure or kernel opens the possibility of functional central limit theorems in settings where the plan is treated as an object in a function space.
  • The result suggests concrete sample-size guidelines for colocalization studies once the Lipschitz or smoothness constants of the cost are known.

Load-bearing premise

The functionals of interest, including the colocalization process, must be Hadamard differentiable at the population EOT plan with respect to the underlying marginal probability measures.

What would settle it

A simulation study with increasing sample size that shows the empirical colocalization process fails to approach the predicted Gaussian limit under the paper's stated regularity conditions on the cost function and entropy parameter would falsify the convergence claim.

Figures

Figures reproduced from arXiv: 2604.12940 by Axel Munk, Luis-Alberto Rodr\'iguez, Santiago Arenas-Velilla.

Figure 1
Figure 1. Figure 1: Top: STED nanoscopy images of Tom20 and Mic60: Section of size 128 × 128 pixels generated using data from the Optimal Transport Colocalization (OTC) dataset in Tameling and Naas, 2021. Bottom: Confidence bands: the dashed line correspond to the EOT curve between the protein assemblies in the above images (with regularization parameter λ = 0.001), with uniform bootstrap confidence bands at level 1−α, for α … view at source ↗
Figure 2
Figure 2. Figure 2: Left: Scenario I: Gaussian vs Gaussian Mixture. In blue + the source sam￾ple from the Gaussian distribution and in red × the target sample from the Gaussian mixture. Observe that the sample mixture form three clusters, whose means lie at different distances from the mean of the source distribution. The gray lines repre￾sent the EOT plan (with regularization parameter λ = 0.001) between the samples. Right: … view at source ↗
Figure 3
Figure 3. Figure 3: Bootstrap accuracy of the EOT limit law for scenario I. Q-Q plots of the bootstrap distribution of the uniform norm of the empirical EOT curve for three sample sizes n = 100, 500, 1000 and bootstrap replications B = 1000. The underlying true EOT curve is obtained from a sample of size 20 000, which serves as an approx￾imation of the ground truth. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: EOT-Confidence bands for scenario I: The dashed line corresponds to the colocalization curve (which serves as ground truth) between the Gaussian and the Gaussian mixture distributions (with sample size 20 000 and regularization parameter λ = 0.001). The dotted lines correspond to the EOT curves obtained from n = 100, 500 and 1000 observations, shown with uniform bootstrap confidence bands at level 1 − α. H… view at source ↗
Figure 5
Figure 5. Figure 5: Point-cloud, von Mises-Fisher vs Mixture: In blue the source sample from the von Misses-Fisher distribution and in orange the target sample from the von Mises mixture. Observe the three components of the mixture, each centered at a different distance from the mean of the source sample distribution. 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Quantiles bootstrap 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Quantiles sample n=100, B = 1000 … view at source ↗
Figure 6
Figure 6. Figure 6: Bootstrap accuracy of the EOT limit law for scenario II: Q-Q plots of the bootstrap distribution of the uniform norm of the EOT for three sample sizes n = 100, 500, 1000 and bootstrap replications B = 1000. The underlying true EOT curve is approximated by a sample of size 20 000. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: EOT-confidence bands for scenario II: the dashed line correspond to the EOT curve (approximation of the ground truth) between the von Mises-Fisher and the von Mises-Fisher mixture distributions (with sample size 20 000 and regularization parameter λ = 0.001), the dotted line correspond to the empirical EOT curve with uniform bootstrap confidence bands at level 1 − α, with α = 0.05, based on the n out of n … view at source ↗
Figure 8
Figure 8. Figure 8: Influence of the regularization parameter λ. The dashed lines correspond to the EOT curves between Tom20, Mic60 computed from the image section of size 128 × 128 (see [PITH_FULL_IMAGE:figures/full_fig_p023_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: STED nanoscopy images of Tom20 and Mic60. Top: Complete image of size 472 × 1208 pixels from the Optimal Transport Colocalization (OTC) data set Tameling and Naas (2021). For visualization, pixel intensities were scaled by factors of 5 (Tom20) and 2.5 (Mic60) to enhance brightness and contrast. Bottom: Zoom in (128 × 128 pixels) at the grey rectangles which correspond to the sections in Figures 1 and 8. Ac… view at source ↗
Figure 10
Figure 10. Figure 10: Colocalization curve. EOT curve φ(µbL,n, νbL,n) between the full Tom20 and Mic60 images (see [PITH_FULL_IMAGE:figures/full_fig_p026_10.png] view at source ↗
read the original abstract

Recently, the statistical properties of empirical Entropic Optimal Transport (EOT) have attracted great interest, as this quantity has been shown to be useful for complex data analysis, among other reasons due to its computational efficiency. In several applications, it has been observed that the EOT plan provides valuable information beyond just the optimal value. For example, in cell biology, colocalization analysis based on the EOT plan has been introduced as a measure for quantification of spatial proximity of different protein assemblies. Despite recent progress in the analysis of its risk properties, a precise understanding of its statistical fluctuations to make it accessible for inference remains elusive to a large extent. In this paper, we derive asymptotic weak convergence result for a large class of functionals of the EOT plan, in which the colocalization process is included. The proof is based on Hadamard differentiability and the extended delta method. As an application, we obtain uniform confidence bands for colocalization curves and bootstrap consistency. Our theory is supported by simulation studies and is illustrated by real world data analysis from mitochondrial protein colocalization.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript establishes asymptotic weak convergence results for a broad class of functionals of the empirical entropic optimal transport (EOT) plan, including the colocalization process, by applying Hadamard differentiability of the map from marginal distributions to the EOT plan (or functionals thereof) together with the extended delta method. These results are used to derive uniform confidence bands for colocalization curves and bootstrap consistency. The theory is illustrated with simulation studies and a real-data example involving mitochondrial protein colocalization.

Significance. If the Hadamard differentiability conditions hold under the paper's regularity assumptions on the cost, entropy parameter, and marginals, the work would supply a rigorous distributional limit theory for EOT-based statistics, moving beyond existing risk bounds to enable valid inference. The explicit treatment of the colocalization functional is a practical strength given its use in cell biology, and the combination of general functional-analytic arguments with empirical validation is a positive feature.

major comments (2)
  1. [Section 3, Theorem 3.1] Section 3, Theorem 3.1 and Assumption 3.2: The claim that the colocalization functional is Hadamard differentiable at the population EOT plan relies on sufficient smoothness and strict convexity of the cost together with a fixed positive entropy parameter; however, the verification that the directional derivative remains continuous in the weak topology (or Wasserstein metric) when the entropy parameter is small but positive is not carried out explicitly, which is load-bearing for the subsequent application of the extended delta method to obtain the limiting process.
  2. [Section 5, Equation (5.2)] Section 5, Equation (5.2) and the mitochondrial data application: The uniform confidence bands and bootstrap consistency for the colocalization curves are derived directly from the weak convergence result, but no diagnostic is provided to confirm that the entropy parameter and cost function satisfy the differentiability conditions in the specific mitochondrial protein setting; without this, the coverage guarantees for the real-data illustration remain conditional on unverified assumptions.
minor comments (2)
  1. [Introduction] The definition of the colocalization process in the introduction could be accompanied by a brief reminder of its relation to the EOT plan to improve readability for readers outside optimal transport.
  2. [Simulation studies] In the simulation section, several plots of empirical coverage lack explicit indication of the nominal level (e.g., 95%) on the y-axis, which makes visual assessment slightly harder.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for their careful reading and constructive feedback on our manuscript. The comments help clarify the presentation of the Hadamard differentiability arguments and strengthen the real-data application. We address each major comment point by point below and will incorporate revisions to improve rigor.

read point-by-point responses

  1. Referee: [Section 3, Theorem 3.1] Section 3, Theorem 3.1 and Assumption 3.2: The claim that the colocalization functional is Hadamard differentiable at the population EOT plan relies on sufficient smoothness and strict convexity of the cost together with a fixed positive entropy parameter; however, the verification that the directional derivative remains continuous in the weak topology (or Wasserstein metric) when the entropy parameter is small but positive is not carried out explicitly, which is load-bearing for the subsequent application of the extended delta method to obtain the limiting process.

    Authors: We thank the referee for this precise observation. Under Assumption 3.2, the smoothness and strict convexity of the cost together with fixed ε > 0 ensure Hadamard differentiability of the EOT map via the dual formulation and the regularizing effect of the entropy term. The continuity of the directional derivative in the weak topology follows from the compactness of the space of measures and the fact that the entropy prevents degeneracy for any fixed positive ε (including small values). Nevertheless, we acknowledge that an explicit verification of this continuity was not isolated as a separate step. In the revised manuscript we will add a short lemma (or remark) immediately after the statement of Theorem 3.1 that confirms the continuity of the derivative map under weak convergence of the marginals, thereby making the invocation of the extended delta method fully self-contained. revision: yes

  2. Referee: [Section 5, Equation (5.2)] Section 5, Equation (5.2) and the mitochondrial data application: The uniform confidence bands and bootstrap consistency for the colocalization curves are derived directly from the weak convergence result, but no diagnostic is provided to confirm that the entropy parameter and cost function satisfy the differentiability conditions in the specific mitochondrial protein setting; without this, the coverage guarantees for the real-data illustration remain conditional on unverified assumptions.

    Authors: We agree that an explicit diagnostic for the mitochondrial example would make the coverage guarantees more convincing. In that application we used the squared Euclidean cost (which is C^∞ and strictly convex) and a fixed positive entropy parameter chosen for numerical stability. To address the referee’s concern, the revised Section 5 will contain a brief paragraph verifying that these choices satisfy Assumption 3.2 and the conditions of Theorem 3.1. We will also report a small sensitivity check over a range of nearby ε values to illustrate that the resulting confidence bands remain stable, thereby grounding the real-data illustration in the verified hypotheses. revision: yes

Circularity Check

0 steps flagged

No circularity; applies Hadamard differentiability and delta method to EOT

full rationale

The derivation establishes weak convergence of functionals of the empirical EOT plan (including colocalization) by invoking Hadamard differentiability of the map from marginals to the plan/functional, followed by the extended delta method. These are standard external functional-analytic tools applied to the EOT setting under explicitly stated regularity assumptions on the cost, entropy parameter, and marginals. No self-definitional steps, no fitted parameters renamed as predictions, and no load-bearing self-citations appear in the abstract or described proof strategy. The central result does not reduce to its inputs by construction; it is an application of general theorems whose validity is independent of the present paper.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review yields no explicit list of free parameters or invented entities; the derivation rests on standard domain assumptions from empirical process theory and optimal transport that are not enumerated here.

axioms (1)
  • domain assumption The map from probability measures to the EOT plan is Hadamard differentiable at the population level for the functionals of interest
    Invoked to apply the extended delta method to obtain the limiting distribution of the empirical functionals.

pith-pipeline@v0.9.0 · 5490 in / 1354 out tokens · 68155 ms · 2026-05-10T13:56:20.408867+00:00 · methodology

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Reference graph

Works this paper leans on

7 extracted references · 4 canonical work pages

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    Denote, with abuse of notation, byξthe operator onC b(X)× C b(X)defined byξ(f, g)(x, y), where x, y∈ X

    Letf, g∈ C b(X)with∥f∥ ∞,∥g∥ ∞ ≤ ∥c∥ ∞ and R X gdν= 0. Denote, with abuse of notation, byξthe operator onC b(X)× C b(X)defined byξ(f, g)(x, y), where x, y∈ X. Thenξis Fr´ echet differentiable with derivativeDξ (f,g) hf , hg (x, y) = ξ(f, g)(x, y) hf (x)+hg(y) λ

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    Letµ, ν∈ P(X)

    Further, assume (LCm). Letµ, ν∈ P(X). Then, the duality mappingψis bounded directional differentiable tangentially to(−µ+P(X)) ×(−ν+P(X))with derivativeψ ′ (µ,ν) =−∂ 2Ψ−1 (µ,ν,ψ(µ,ν)) ∂1Ψ(µ,ν,ψ(µ,ν)). DefineΞ(µ, ν) =ξ f λ µ,ν, gλ µ,ν .Ξis bounded directional differentiable tangen- tially to(−µ+P(X))×(−ν+P(X))with derivative Ξ(µ,ν) (hµ, hν) (x, y) =ξ f λ µ...

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    As it was pointed in Section 1, the mapψis well defined by existence and uniqueness of solution theorem for the entropic optimal transport problem (1) (see Nutz, 2021)

    To prove this result the implicit mapping theorem in Banach spaces is to be used (see Zeidler, 2012, Section 4.8). As it was pointed in Section 1, the mapψis well defined by existence and uniqueness of solution theorem for the entropic optimal transport problem (1) (see Nutz, 2021). More precisely, for fixedµ, ν∈ P(X) there exists a pair f λ µ,ν, gλ µ,ν ∈...

    2012