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arxiv: 2606.10767 · v1 · pith:U5UTVMFUnew · submitted 2026-06-09 · 📊 stat.ME

Two-Sample Homogeneity Test via Entropic Optimal Transport

Yiming Ma , Hang Liu , Weiwei Zhuang This is my paper

Pith reviewed 2026-06-27 12:26 UTC · model grok-4.3

classification 📊 stat.ME
keywords two-sample testentropic optimal transporthomogeneity testingfunctional central limit theoremmultiplier bootstrapmap discrepancyconsistencylocal power
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The pith

The squared L2 distance between empirical entropic optimal transport maps from a uniform reference provides a consistent test for equality of two distributions.

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

This paper shows that the squared L2 distance between empirical entropic optimal transport maps, each obtained by pushing forward a fixed uniform distribution on the unit ball, can serve as a test statistic for whether two unknown distributions are identical. For a fixed regularization parameter the population maps differ if and only if the distributions differ, and under the null the centered empirical map process converges to a Gaussian process whose quadratic form gives the null limit. A multiplier bootstrap approximates the distribution of that quadratic form, the test is consistent against fixed alternatives, and its local power is characterized for contiguous alternatives. The approach also supplies map-based diagnostics that reveal the nature of any detected difference.

Core claim

The authors establish that the population map discrepancy is identifiable, derive the functional CLT for the empirical map difference under the null yielding a Gaussian quadratic-form limit, prove consistency against fixed alternatives, characterize local asymptotic power under contiguous alternatives, and validate a weighted multiplier bootstrap for the non-pivotal null distribution.

What carries the argument

The entropic optimal transport map from the uniform law on the unit ball to each target distribution; the squared L2 norm of the difference between two such maps forms the test statistic.

Load-bearing premise

The entropic regularization parameter is fixed in advance and the reference measure is the uniform distribution on the unit ball, with the data distributions admitting well-behaved EOT maps.

What would settle it

Empirical rejection rates under the null that deviate substantially from the nominal level, or failure to reject with high probability when the two distributions are known to differ by a fixed amount.

Figures

Figures reproduced from arXiv: 2606.10767 by Hang Liu, Weiwei Zhuang, Yiming Ma.

Figure 1
Figure 1. Figure 1: Illustration of the proposed common-reference map comparison. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Empirical power under fixed alternatives. The nominal level is [PITH_FULL_IMAGE:figures/full_fig_p022_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Empirical rejection rates under location alternatives [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Visualization of the estimated EOT-map discrepancy in dimension [PITH_FULL_IMAGE:figures/full_fig_p027_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Spatial distributions of Citi Bike starting locations. Left: member [PITH_FULL_IMAGE:figures/full_fig_p029_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Empirical EOT-map discrepancy on the source unit disk. The color [PITH_FULL_IMAGE:figures/full_fig_p031_6.png] view at source ↗
read the original abstract

This paper proposes a two-sample homogeneity test based on entropic optimal transport (EOT) maps from a common reference distribution -- the uniform law on the unit ball. The test statistic is the squared $L^2$-distance between the two empirical EOT maps. For fixed entropic regularization parameter, we prove that the population map discrepancy is identifiable, derive a functional central limit theorem for the empirical map difference under the null, and establish the Gaussian quadratic-form null limit. We also prove consistency against fixed alternatives and characterize local asymptotic power under contiguous alternatives. A weighted multiplier bootstrap is proposed to calibrate the non-pivotal null distribution, and its validity is established. Extensive simulations demonstrate that the proposed EOT-map test has reliable finite-sample size control and exhibits competitive power compared with other existing methods. The method is particularly powerful for location alternatives and, beyond a single scalar discrepancy, it provides additional diagnostic information on how the two distributions differ. Finally, a real data application concludes the paper.

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 / 1 minor

Summary. The paper proposes a two-sample homogeneity test using the squared L² distance between empirical entropic optimal transport (EOT) maps from a fixed common reference (uniform on the unit ball). For fixed regularization parameter, it claims to prove identifiability of the population map discrepancy, a functional CLT for the empirical map difference under the null yielding a Gaussian quadratic-form limit, consistency against fixed alternatives, local asymptotic power under contiguous alternatives, and validity of a weighted multiplier bootstrap for the non-pivotal limit. Simulations and a real-data example are included.

Significance. If the regularity conditions hold, the method supplies both a calibrated test statistic and diagnostic information on distributional differences beyond a scalar p-value, with competitive power for location shifts. The combination of identifiability, FCLT, bootstrap validity, and local-power characterization is a substantive contribution to nonparametric two-sample testing.

major comments (2)
  1. [FCLT theorem statement and proof] The functional CLT (and hence the Gaussian quadratic-form null limit and bootstrap validity) rests on tightness of the centered empirical EOT-map process in the relevant function space. The manuscript must explicitly state the moment/support conditions on P and Q (relative to the fixed uniform reference on the unit ball) that guarantee this tightness; without them the central limit theorem does not follow from standard empirical-process arguments.
  2. [Identifiability and consistency sections] The identifiability claim and consistency result are stated for the population map discrepancy under the fixed reference measure. The paper should clarify whether the unit-ball support of the reference is essential or whether the results extend to other compactly supported references without altering the test statistic's form.
minor comments (1)
  1. [Notation and bootstrap section] Notation for the EOT map functional and the precise definition of the weighted multiplier bootstrap weights should be introduced earlier and used consistently.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will revise the paper accordingly to strengthen the presentation.

read point-by-point responses

  1. Referee: [FCLT theorem statement and proof] The functional CLT (and hence the Gaussian quadratic-form null limit and bootstrap validity) rests on tightness of the centered empirical EOT-map process in the relevant function space. The manuscript must explicitly state the moment/support conditions on P and Q (relative to the fixed uniform reference on the unit ball) that guarantee this tightness; without them the central limit theorem does not follow from standard empirical-process arguments.

    Authors: We agree that the conditions guaranteeing tightness of the empirical EOT-map process must be stated explicitly for the FCLT to be fully rigorous. The current manuscript assumes P and Q admit densities with respect to Lebesgue measure and possess finite moments of sufficiently high order (to control the Lipschitz constants of the EOT potentials), but these are not collected in a single assumption. In the revision we will add an explicit Assumption (new Assumption 2.2) stating that P and Q are supported on a fixed compact convex set containing the unit ball in its interior and have finite (2+δ)-moments for some δ>0; we will then verify in the proof of Theorem 3.1 that these conditions imply the required entropy-integrability condition for tightness in the Hölder space used for the functional CLT. revision: yes

  2. Referee: [Identifiability and consistency sections] The identifiability claim and consistency result are stated for the population map discrepancy under the fixed reference measure. The paper should clarify whether the unit-ball support of the reference is essential or whether the results extend to other compactly supported references without altering the test statistic's form.

    Authors: The unit ball is selected for computational convenience and to ensure an explicit characterization of the reference measure, but it is not essential to the theoretical results. Identifiability of the map discrepancy follows from the strict convexity of the entropic transport cost and the uniqueness of the Brenier potential for any reference measure that is absolutely continuous with respect to Lebesgue measure on a compact convex set with nonempty interior; the same argument applies verbatim to consistency. We will insert a short remark after Definition 2.1 clarifying that all results in Sections 3 and 4 continue to hold, with no change to the form of the test statistic, when the reference is replaced by any other fixed compactly supported probability measure satisfying the same absolute-continuity condition. revision: yes

Circularity Check

0 steps flagged

No circularity: standard empirical-process arguments applied to EOT map functional

full rationale

The paper states that for fixed entropic regularization it proves identifiability of the population map discrepancy, derives a functional CLT for the empirical map difference under the null, obtains a Gaussian quadratic-form limit, proves consistency, and characterizes local power, all under the maintained assumption that the data-generating distributions admit well-defined EOT maps from the fixed uniform reference on the unit ball. These are presented as consequences of standard tightness and convergence conditions in the relevant function space rather than as quantities fitted or defined in terms of the target test statistic. No self-definitional loop, fitted-input prediction, or load-bearing self-citation chain appears in the derivation chain; the central claims rest on external empirical-process theory applied to the EOT functional.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The development relies on standard assumptions from optimal transport and empirical process theory; the regularization parameter is treated as fixed rather than estimated.

free parameters (1)
  • entropic regularization parameter
    Held fixed throughout the asymptotic analysis; its specific value is not derived from data but chosen by the user.
axioms (2)
  • domain assumption The EOT map functional satisfies the conditions for a functional central limit theorem under the null (tightness, finite-dimensional convergence).
    Invoked to obtain the Gaussian quadratic-form null limit.
  • domain assumption The population map discrepancy is identifiable when the two distributions differ.
    Stated as proved for fixed regularization.

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discussion (0)

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