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

arXiv: https://arxiv.org/abs/2512.13997 · HTML: https://mlsanity.com/paper/2512.13997/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 50 references that are in the map.
- First posted on arXiv 16 Dec 2025.

## Builds on (most cited first, 19 of 19)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Do ImageNet Classifiers Generalize to ImageNet?](https://mlsanity.com/paper/1902.10811/) (2019)
- [Demystifying MMD GANs](https://mlsanity.com/paper/1801.01401/) (2018)
- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Rethinking FID: Towards a Better Evaluation Metric for Image Generation](https://mlsanity.com/paper/2401.09603/) (2023)
- [Revisiting Classifier Two-Sample Tests](https://mlsanity.com/paper/1610.06545/) (2016)
- [Generative Moment Matching Networks](https://mlsanity.com/paper/1502.02761/) (2015)
- [Training generative neural networks via Maximum Mean Discrepancy optimization](https://mlsanity.com/paper/1505.03906/) (2015)
- [Learning Deep Kernels for Non-Parametric Two-Sample Tests](https://mlsanity.com/paper/2002.09116/) (2020)
- [Generative Models and Model Criticism via Optimized Maximum Mean Discrepancy](https://mlsanity.com/paper/1611.04488/) (2016)
- [MMD Aggregated Two-Sample Test](https://mlsanity.com/paper/2110.15073/) (2021)
- [Fast Two-Sample Testing with Analytic Representations of Probability Measures](https://mlsanity.com/paper/1506.04725/) (2015)
- [Classification accuracy as a proxy for two sample testing](https://mlsanity.com/paper/1602.02210/) (2016)
- [A Witness Two-Sample Test](https://mlsanity.com/paper/2102.05573/) (2021)
- [A Permutation-free Kernel Two-Sample Test](https://mlsanity.com/paper/2211.14908/) (2022)
- [AutoML Two-Sample Test](https://mlsanity.com/paper/2206.08843/) (2022)
- [MMD-B-Fair: Learning Fair Representations with Statistical Testing](https://mlsanity.com/paper/2211.07907/) (2022)
- [Unbiased estimators for the variance of MMD estimators](https://mlsanity.com/paper/1906.02104/) (2019)
- [Product-form estimators: exploiting independence to scale up Monte Carlo](https://mlsanity.com/paper/2102.11575/) (2021)

## Built on by (most cited first, 1 of 1)

- [Finite-Sample Unbiased Variance of MMD under Unbalanced Sampling: Exact Estimation and Quasi-Linear Computation](https://mlsanity.com/paper/2601.13874/) (2026)

## Abstract (arXiv)

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.