📊 statistics

Two-Sample Homogeneity Test via Entropic Optimal Transport

This paper introduces a two-sample homogeneity test based on the squared $L^2$ -distance between entropic optimal transport maps from a common reference distribution, establishing its theoretical properties, proposing a weighted multiplier bootstrap for calibration, and demonstrating its competitive power and diagnostic capabilities through simulations and real data applications.

Original authors: Yiming Ma, Hang Liu, Weiwei Zhuang

Published 2026-06-10

📖 5 min read🧠 Deep dive

CC0 1.0

Original authors: Yiming Ma, Hang Liu, Weiwei Zhuang

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to figure out if two groups of people are actually the same crowd, just shuffled around, or if they are fundamentally different groups.

In statistics, this is called a Two-Sample Homogeneity Test. Usually, you have a pile of data from Group A and a pile from Group B. You want to know: Are these two piles coming from the same underlying reality, or are they different?

Most existing methods try to answer this by squashing the entire complex shape of the data down into a single number (like a distance score). If the number is big, they say, "Different!" If it's small, they say, "Same!" But this is like judging two different cities only by their average temperature. You lose all the interesting details about how they differ.

This paper proposes a new, more detailed way to solve this mystery using a concept called Entropic Optimal Transport (EOT). Here is how it works, explained simply:

1. The "Universal Map" Analogy

Instead of comparing Group A directly to Group B, the authors introduce a third party: a Reference Map.

The Setup: Imagine a perfect, empty, round city called "Unit Ball City." It has a standard grid layout.
The Transformation: The authors create a "transport map" for Group A. This map shows exactly how to move every resident of Unit Ball City to their new home in Group A's neighborhood. It's like a GPS route for every single person.
The Second Map: They do the exact same thing for Group B. They create a second GPS map showing how to move residents from the same Unit Ball City to Group B's neighborhood.

Now, instead of comparing two messy neighborhoods, the authors are comparing two GPS maps that start from the exact same place.

2. The "Difference Vector"

Once they have these two maps, they look at the difference between them.

At any specific point in Unit Ball City, the Map for Group A might say, "Go 5 miles North."
The Map for Group B might say, "Go 5 miles East."
The "discrepancy" is the difference between those two directions.

The test statistic is essentially the total squared distance between these two GPS maps across the whole city. If the maps are identical, the groups are the same. If the maps point in wildly different directions, the groups are different.

3. Why This is Better (The "Diagnostic" Power)

The paper claims this method has two superpowers:

It's a Detective, Not Just a Judge: Most tests just give you a "Yes/No" answer. This method gives you a visual map of the difference.
- If Group A is just Group B shifted slightly to the right, the map shows a uniform arrow pointing right.
- If Group B is more spread out (larger variance), the map shows arrows fanning outward like a starburst.
- If the relationship between variables is twisted, the map shows a shearing or swirling pattern.
- Analogy: It's like looking at a weather map. A standard test just tells you "It's raining." This test shows you where the rain is, how hard it's falling, and which way the wind is blowing.
It Handles Complexity Well: The authors prove mathematically that if the maps are different, the groups must be different (identifiability). They also show that their method is very good at detecting when groups are slightly different (high power), especially when the difference is a shift in location (like the mean).

4. The "Bootstrap" Safety Net

Since calculating the exact mathematical probability of these differences is incredibly hard (like trying to predict the exact path of every raindrop), the authors use a clever trick called a Weighted Multiplier Bootstrap.

The Analogy: Imagine you have a deck of cards representing your data. To see if your result is a fluke, you shuffle the deck, add some random noise (multipliers), and re-run the test thousands of times.
By doing this, they create a "confidence zone" to decide if the difference they found is real or just random noise. The paper proves this trick works reliably.

5. Real-World Test: The Bike Riders

To prove it works, they tested it on Citi Bike data from Jersey City.

The Question: Do "Member" riders and "Casual" riders start their trips from the same places?
The Result: The test said "No, they are different."
The Bonus: The map didn't just say "No." It showed where the difference was. It revealed that while both groups ride in the downtown area, the intensity and specific starting spots differed in a way that a simple "average" test might have missed. It showed that Members and Casuals have distinct usage patterns.

Summary

This paper introduces a new statistical tool that compares two groups by mapping them both against a common, standard reference.

Old Way: Compare two messy piles of data with a ruler (get one number).
New Way: Compare two detailed GPS maps of how to get to those piles (get a visual, directional map of the differences).

The authors show that this new way is mathematically sound, works well with computers, and gives you much richer information about how two groups of data are different, not just that they are different.

Technical Summary: Two-Sample Homogeneity Test via Entropic Optimal Transport

Problem Statement
The paper addresses the fundamental statistical problem of two-sample homogeneity testing in multivariate settings ( $d \ge 1$ ). Given two independent samples from unknown distributions $P$ and $Q$ on $\mathbb{R}^d$ , the goal is to test the null hypothesis $H_0: P = Q$ against $H_1: P \neq Q$ . While univariate testing benefits from canonical ordering (e.g., quantile functions), multivariate testing lacks a natural order, making it difficult to construct distributional representations that preserve geometric structure. Existing methods, such as energy distances, kernel-based Maximum Mean Discrepancy (MMD), and graph-based statistics, typically summarize distributional differences into a single scalar value. This aggregation often discards map-level information regarding how one distribution is organized relative to another or a common reference. Furthermore, inference based on empirical Wasserstein distances in multivariate settings is often computationally expensive and suffers from nonstandard asymptotics.

Methodology
The authors propose a homogeneity test based on Entropic Optimal Transport (EOT) maps from a common reference distribution. The methodology proceeds as follows:

Common Reference Construction: Let $U_d$ be the uniform distribution on the unit ball $B_d \subset \mathbb{R}^d$ . For any target distribution $P$ , the method defines an entropic transport map $T_{\varepsilon, P}: B_d \to \mathbb{R}^d$ . This map is the barycentric projection of the unique entropic optimal coupling $\pi_{\varepsilon, P}$ between $U_d$ and $P$ , minimizing the cost $\int \frac{1}{2}\|u-x\|^2 d\pi + \varepsilon \text{KL}(\pi \| U_d \otimes P)$ .
Empirical Estimation: Given samples $X_1, \dots, X_n \sim P$ and $Y_1, \dots, Y_m \sim Q$ , the population maps are estimated by their empirical counterparts $\hat{T}_{\varepsilon, n}$ and $\hat{T}_{\varepsilon, m}$ , where the source distribution remains the fixed $U_d$ (approximated numerically by a reference sample) and the target is replaced by the empirical measures $P_n$ and $Q_m$ .
Test Statistic: The test statistic $T_{n,m}$ is the scaled squared $L^2$ -distance between the two empirical maps over the reference domain:
$T_{n,m} = \frac{nm}{n+m} \int_{B_d} \| \hat{T}_{\varepsilon, n}(u) - \hat{T}_{\varepsilon, m}(u) \|^2 dU_d(u).$
Calibration: Since the null distribution of $T_{n,m}$ depends on the unknown covariance structure of the limiting Gaussian process (non-pivotal), the authors propose a weighted multiplier bootstrap. This procedure reweights the empirical target measures separately while keeping the source fixed, generating a conditional distribution to approximate critical values.

Key Contributions
The paper establishes the theoretical and practical foundations for this map-based testing framework:

Identifiability: Under compact support and fixed regularization assumptions, the authors prove that the population EOT map uniquely determines the underlying distribution. Specifically, $T_{\varepsilon, P} = T_{\varepsilon, Q}$ $U_d$ -almost everywhere if and only if $P = Q$ . This ensures the population discrepancy vanishes precisely under the null.
Asymptotic Theory:
- Null Limit: A functional central limit theorem (FCLT) is derived for the empirical map difference under $H_0$ . The statistic converges to a Gaussian quadratic form $\sum \omega_k Z_k^2$ , where $\omega_k$ are eigenvalues of the covariance operator of the limiting Gaussian field.
- Consistency: The test is shown to be consistent against fixed alternatives ( $P \neq Q$ ), with the statistic diverging to infinity.
- Local Power: The asymptotic power under contiguous local alternatives is characterized, showing the test is sensitive to differences in the tangent directions of the distributions.
Bootstrap Validity: The validity of the weighted multiplier bootstrap is established, proving that the conditional distribution of the bootstrap statistic consistently estimates the null limit distribution.
Diagnostic Capability: Unlike scalar discrepancy tests, the method provides an estimated vector field ( $\hat{T}_{\varepsilon, n} - \hat{T}_{\varepsilon, m}$ ) that visualizes the direction and nature of distributional differences (e.g., translation, radial expansion, shear).

Results
Extensive numerical experiments and a real-data application support the theoretical findings:

Finite-Sample Calibration: The weighted multiplier bootstrap provides reliable size control across various null distributions (Gaussian, Student-t, Gaussian mixtures) and dimensions ( $d=2, 5, 10$ ), with empirical rejection rates close to the nominal level (0.05).
Power: The test demonstrates competitive power against existing methods (Energy, MMD, Sinkhorn divergence, k-NN, Wasserstein, OT-rank). It exhibits particularly strong performance under location alternatives, where the coherent directional displacement between maps is easily detected. It also shows robust power under scale, correlation, and mixture alternatives, though performance varies relative to other methods depending on the specific alternative and dimension.
Computational Efficiency: While the EOT-map test is slower than Energy or MMD, it is faster than permutation-calibrated Sinkhorn divergence. The computational cost is tunable via the number of reference points $N$ ; moderate $N$ values yield stable power and size while significantly reducing runtime.
Real-Data Application: Applied to Citi Bike trip data (member vs. casual riders), the test detected significant differences in starting location distributions. The map-level visualization revealed that the discrepancy was not merely a global shift but involved complex spatial variations, with coordinate-wise decomposition showing contributions from both longitude and latitude.

Significance and Claims
The paper positions the proposed EOT-map test as a competitive and interpretable complement to existing scalar discrepancy tests, rather than a uniformly dominant replacement. Its primary significance lies in:

Geometric Interpretability: It retains directional and geometric information about distributional changes, allowing researchers to diagnose how distributions differ (e.g., via vector field visualization) rather than just if they differ.
Theoretical Rigor: It provides a rigorous limit theory for EOT maps under fixed regularization, overcoming the nonstandard asymptotics often associated with unregularized Wasserstein distances.
Practical Utility: The method offers a practical balance between statistical power, computational tractability (via Sinkhorn algorithms and bootstrap), and diagnostic insight, making it suitable for applications where understanding the structure of distributional shifts is as important as the test decision itself.