Unified Unbiased Variance Estimation for Maximum Mean Discrepancy: Robust Finite-Sample Performance with Imbalanced Data and Exact Acceleration under Null and Alternative Hypotheses

2026-01-20

AI Summary

This work addresses the inconsistent performance of existing variance estimators for the Maximum Mean Discrepancy (MMD) two-sample test under varying conditions—specifically, across the null and alternative hypotheses as well as balanced and unbalanced sample settings—and the absence of a unified framework. By leveraging the U-statistic representation and Hoeffding decomposition, the authors establish the first unified, unbiased variance estimation framework for MMD that encompasses all such hypothesis and sampling configurations. Furthermore, for the one-dimensional Laplacian kernel, they develop an exact accelerated algorithm that reduces computational complexity from O(n²) to O(n log n). The proposed method demonstrates robustness in finite samples, significantly enhancing both statistical inference accuracy and computational efficiency.

Technology Category

• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

The maximum mean discrepancy (MMD) is a kernel-based nonparametric statistic for two-sample testing, whose inferential accuracy depends critically on variance characterization. Existing work provides various finite-sample estimators of the MMD variance, often differing under the null and alternative hypotheses and across balanced or imbalanced sampling schemes. In this paper, we study the variance of the MMD statistic through its U-statistic representation and Hoeffding decomposition, and establish a unified finite-sample characterization covering different hypotheses and sample configurations. Building on this analysis, we propose an exact acceleration method for the univariate case under the Laplacian kernel, which reduces the overall computational complexity from $\mathcal O(n^2)$ to $\mathcal O(n \log n)$.

Problem

• Maximum Mean Discrepancy
• Variance Estimation
• Two-sample Testing
• Imbalanced Data
• U-statistic

Innovation

• MMD variance estimation
• U-statistic
• Hoeffding decomposition
• computational acceleration
• imbalanced data