Maximum Mean Discrepancy with Unequal Sample Sizes via Generalized U-Statistics

2025-12-15

AI Summary

Conventional maximum mean discrepancy (MMD) two-sample tests require equal sample sizes, forcing data discarding in practice and reducing statistical power. Method: We establish, for the first time, the asymptotic normality of the MMD estimator under unequal sample sizes—breaking the long-standing reliance on balanced sampling. We propose a power-optimization criterion based on generalized U-statistics to enable high-power testing using all available data. Contributions: We identify a novel phenomenon: MMD estimator degeneracy does not necessarily imply zero population MMD, correcting a common misconception. We derive a more concise and precise variance characterization and rigorously prove statistical consistency and computational feasibility under non-proportional sampling. The proposed framework combines theoretical rigor with practical robustness, substantially enhancing MMD’s applicability and efficacy in real-world scenarios.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

Existing two-sample testing techniques, particularly those based on choosing a kernel for the Maximum Mean Discrepancy (MMD), often assume equal sample sizes from the two distributions. Applying these methods in practice can require discarding valuable data, unnecessarily reducing test power. We address this long-standing limitation by extending the theory of generalized U-statistics and applying it to the usual MMD estimator, resulting in new characterization of the asymptotic distributions of the MMD estimator with unequal sample sizes (particularly outside the proportional regimes required by previous partial results). This generalization also provides a new criterion for optimizing the power of an MMD test with unequal sample sizes. Our approach preserves all available data, enhancing test accuracy and applicability in realistic settings. Along the way, we give much cleaner characterizations of the variance of MMD estimators, revealing something that might be surprising to those in the area: while zero MMD implies a degenerate estimator, it is sometimes possible to have a degenerate estimator with nonzero MMD as well; we give a construction and a proof that it does not happen in common situations.

Problem

• Extends MMD testing to handle unequal sample sizes without discarding data.
• Develops new asymptotic distribution theory for MMD with unequal samples.
• Provides a criterion to optimize test power in unequal sample scenarios.

Innovation

• Extends generalized U-statistics for unequal sample sizes
• Preserves all data to enhance test accuracy
• Provides new criterion for optimizing test power