Core Concepts

The authors develop a projected Wasserstein distance to circumvent the curse of dimensionality in high-dimensional two-sample testing, and provide theoretical guarantees for the finite-sample convergence rate of the proposed distance.

Abstract

The key highlights and insights from the content are:
The authors consider the problem of two-sample testing, where the goal is to determine whether two sets of samples are from the same underlying distribution or not. This is a fundamental problem in statistics and machine learning with applications in areas like anomaly detection, change-point detection, and model criticism.
Classical parametric two-sample tests like Hotelling's test and Student's t-test are not suitable for high-dimensional non-parametric settings. The authors focus on non-parametric two-sample tests based on integral probability metrics (IPMs) like the Wasserstein distance.
The key challenge with using the Wasserstein distance for high-dimensional two-sample testing is the slow convergence rate of the empirical Wasserstein distance, which suffers from the curse of dimensionality.
To address this issue, the authors propose the projected Wasserstein distance, which finds the low-dimensional linear mapping that maximizes the Wasserstein distance between the projected probability distributions.
The authors provide theoretical analysis on the finite-sample convergence rate of the proposed projected Wasserstein distance, showing that it has a milder dependence on the data dimension compared to the standard Wasserstein distance.
Numerical experiments validate the theoretical results and demonstrate that the two-sample test based on the projected Wasserstein distance outperforms existing methods like the Maximum Mean Discrepancy (MMD) test, especially in high dimensions.
The proposed framework also provides an interpretable way to visualize and understand the differences between the two high-dimensional distributions by studying the optimal projection mapping and the projected samples in low dimensions.

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by Jie Wang,Rui... at **arxiv.org** 04-01-2024

Deeper Inquiries

To optimize the choice of the projection dimension k for improving the performance of the two-sample test based on the projected Wasserstein distance, several considerations can be taken into account:
Trade-off between Dimensionality Reduction and Information Loss: Increasing the projection dimension k allows for a more detailed representation of the data in the lower-dimensional space. However, this comes at the cost of potentially losing important information from the original high-dimensional space. By balancing the trade-off between dimensionality reduction and information preservation, the optimal projection dimension can be determined.
Empirical Evaluation: Conducting empirical evaluations with varying values of k can help identify the point at which the performance of the two-sample test stabilizes or starts to degrade. This empirical analysis can provide insights into the impact of different projection dimensions on the test's accuracy and power.
Computational Complexity: The computational complexity of the test increases with higher projection dimensions. Therefore, the optimal choice of k should also consider the computational resources available and the efficiency of the algorithm in handling higher-dimensional projections.
Generalization Bounds: Theoretical analysis, similar to the Rademacher complexity argument used in the paper, can be employed to derive generalization bounds for different projection dimensions. These bounds can guide the selection of k to ensure good performance on unseen data.
By carefully considering these factors and potentially conducting a systematic study or optimization procedure, the choice of the projection dimension k can be optimized to enhance the performance of the two-sample test based on the projected Wasserstein distance.

The projected Wasserstein distance has applications beyond two-sample testing in various high-dimensional statistical inference tasks:
Generative Modeling: In generative modeling, the projected Wasserstein distance can be utilized to compare the distributions of generated samples with real data distributions. By measuring the discrepancy between generated and real data distributions in a lower-dimensional space, the quality of generative models can be assessed more effectively.
Domain Adaptation: In domain adaptation tasks, where the goal is to transfer knowledge from a source domain to a target domain, the projected Wasserstein distance can help quantify the distribution shift between domains. By aligning the distributions in a lower-dimensional space, domain adaptation algorithms can be optimized for better performance.
Statistical Inference: The projected Wasserstein distance can be applied in various statistical inference tasks, such as hypothesis testing, model comparison, and outlier detection. By projecting high-dimensional data onto lower-dimensional spaces and comparing distributions using Wasserstein metrics, more efficient and accurate inference can be achieved.
By extending the use of the projected Wasserstein distance to these applications, researchers can leverage its benefits in addressing high-dimensional challenges across different domains.

Using higher-order Wasserstein distances, such as the 2-Wasserstein distance, in high-dimensional two-sample testing can offer several benefits and insights:
Enhanced Discriminative Power: Higher-order Wasserstein distances capture more complex relationships between distributions compared to lower-order distances. In high-dimensional spaces, where distributions may exhibit intricate structures, using a higher-order Wasserstein distance can provide a more discriminative measure of dissimilarity.
Improved Sensitivity to Distributional Differences: Higher-order Wasserstein distances can be more sensitive to subtle differences between distributions, especially in cases where lower-order distances may not adequately capture the nuances in the data. This increased sensitivity can lead to more accurate and reliable two-sample testing results.
Robustness to Noise: In scenarios where the data is noisy or contains outliers, higher-order Wasserstein distances can offer increased robustness by focusing on the overall structure of the distributions rather than individual data points. This robustness can lead to more stable and reliable statistical inference.
By incorporating higher-order Wasserstein distances into high-dimensional two-sample testing, researchers can potentially uncover deeper insights into the underlying distributional differences and improve the performance of statistical tests in complex data settings.

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