Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Neural Wasserstein Two-Sample Tests

Xiaoyu Hu, Zhenhua Lin

TL;DR

This work develops a learning-assisted procedure based on the projection 1-Wasserstein distance, which it is called the neural Wasserstein test, and establishes the validity and consistency of the proposed test and proves a Berry--Esseen type bound for the Gaussian approximation.

Abstract

The two-sample homogeneity testing problem is fundamental in statistics and becomes particularly challenging in high dimensions, where classical tests can suffer substantial power loss. We develop a learning-assisted procedure based on the projection 1-Wasserstein distance, which we call the neural Wasserstein test. The method is motivated by the observation that there often exists a low-dimensional projection under which the two high-dimensional distributions differ. In practice, we learn the projection directions via manifold optimization and a witness function using deep neural networks. To adapt to unknown projection dimensions and sparsity levels, we aggregate a collection of candidate statistics through a max-type construction, avoiding explicit tuning while potentially improving power. We establish the validity and consistency of the proposed test and prove a Berry--Esseen type bound for the Gaussian approximation. In particular, under the null hypothesis, the aggregated statistic converges to the absolute maximum of a standard Gaussian vector, yielding an asymptotically pivotal (distribution-free) calibration that bypasses resampling. Simulation studies and a real-data example demonstrate the strong finite-sample performance of the proposed method.

Neural Wasserstein Two-Sample Tests

TL;DR

This work develops a learning-assisted procedure based on the projection 1-Wasserstein distance, which it is called the neural Wasserstein test, and establishes the validity and consistency of the proposed test and proves a Berry--Esseen type bound for the Gaussian approximation.

Abstract

The two-sample homogeneity testing problem is fundamental in statistics and becomes particularly challenging in high dimensions, where classical tests can suffer substantial power loss. We develop a learning-assisted procedure based on the projection 1-Wasserstein distance, which we call the neural Wasserstein test. The method is motivated by the observation that there often exists a low-dimensional projection under which the two high-dimensional distributions differ. In practice, we learn the projection directions via manifold optimization and a witness function using deep neural networks. To adapt to unknown projection dimensions and sparsity levels, we aggregate a collection of candidate statistics through a max-type construction, avoiding explicit tuning while potentially improving power. We establish the validity and consistency of the proposed test and prove a Berry--Esseen type bound for the Gaussian approximation. In particular, under the null hypothesis, the aggregated statistic converges to the absolute maximum of a standard Gaussian vector, yielding an asymptotically pivotal (distribution-free) calibration that bypasses resampling. Simulation studies and a real-data example demonstrate the strong finite-sample performance of the proposed method.
Paper Structure (23 sections, 9 theorems, 53 equations, 3 figures, 2 tables, 1 algorithm)

This paper contains 23 sections, 9 theorems, 53 equations, 3 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. A distribution-free test statistic
  3. Projection Wasserstein distance and two-sample tests
  4. Overview of the proposed new two-sample test
  5. An effective algorithm for computing $(\hat{f},\hat{U})$
  6. The test statistic via aggregating multiple projection dimensions and regularization parameters
  7. Limiting null distribution and size analysis
  8. Power analysis
  9. Power enhancement with $\ell_1$ regularization
  10. Estimation and the test statistic
  11. Power analysis
  12. Computational algorithm
  13. Power enhancement with $\ell_0$ regularization
  14. Estimation and the test statistic
  15. Power analysis
  16. ...and 8 more sections

Key Result

Lemma 1

The $PW_k$ is a distance over $\mathcal{P}(\mathbb R^d)$. Also, for $\mu, \nu \in \mathcal{P}(\mathbb R^d)$, there exists a $U_0 \in \mathcal{S}_{d,k}$ such that

Figures (3)

  • Figure 1: Empirical size ($\beta=0$) and power $(\beta>0)$ of various methods in different models with $n_x=n_y=250$ and $d=500$.
  • Figure 2: The differences in mean between GBM and LGG, with error bars indicating the standard errors.
  • Figure 3: The heatmaps of covariance matrices for the subset of variables with the 50 largest univariate 1-Wasserstein distances. Left: GBM; Right: LGG.

Theorems & Definitions (23)

  • Lemma 1: Paty and Cuturi, 2019
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Theorem 1
  • Remark 6
  • Remark 7
  • Lemma 2
  • ...and 13 more