Hypothesis Test and Confidence Analysis With Wasserstein Distance on General Dimension

Area of Science:

• Statistics
• Machine Learning
• Optimal Transport

Background:

• The 1-Wasserstein distance is increasingly used in machine learning due to its beneficial properties.
• Existing statistical inference methods for Wasserstein distance lack generality, especially in multivariate settings.
• The unavailability of limit distributions for empirical Wasserstein distances under general conditions hinders its application.

Purpose of the Study:

• To establish a general framework for statistical inference with the 1-Wasserstein distance.
• To develop hypothesis testing and confidence analysis for the empirical 1-Wasserstein distance in a general multivariate setting.
• To address the limitations of existing methods by providing a novel approximation approach.

Main Methods:

• Development of a novel nonasymptotic Gaussian approximation for the empirical 1-Wasserstein distance.
• Utilizing the Gaussian approximation to construct hypothesis tests and confidence intervals.
• Providing theoretical guarantees for the proposed approximation method.
• Designing an efficient algorithm to implement the approximation.

Main Results:

• A novel nonasymptotic Gaussian approximation for the empirical 1-Wasserstein distance is presented.
• The developed approximation enables robust hypothesis testing and confidence analysis.
• Theoretical guarantees are established for the accuracy and efficiency of the approximation.
• Experimental results numerically validate the performance of the proposed methods.

Conclusions:

• The study introduces a general framework for statistical inference with the 1-Wasserstein distance.
• The novel Gaussian approximation overcomes limitations in existing methods for multivariate settings.
• The developed hypothesis tests and confidence analysis offer reliable tools for machine learning applications.