# Unbiased estimators for the variance of MMD estimators

arXiv: https://arxiv.org/abs/1906.02104 · HTML: https://mlsanity.com/paper/1906.02104/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Jun 2019.

## Built on by (most cited first, 10 of 10)

- [MMD-B-Fair: Learning Fair Representations with Statistical Testing](https://mlsanity.com/paper/2211.07907/) (2022)
- [A Practical Introduction to Kernel Discrepancies: MMD, HSIC & KSD](https://mlsanity.com/paper/2503.04820/) (2025)
- [DUAL: Learning Diverse Kernels for Aggregated Two-sample and Independence Testing](https://mlsanity.com/paper/2510.11140/) (2025)
- [Variance Matters: Improving Domain Adaptation via Stratified Sampling](https://mlsanity.com/paper/2512.05226/) (2025)
- [Learning Representations for Independence Testing](https://mlsanity.com/paper/2409.06890/) (2024)
- [Consecutive Knowledge Meta-Adaptation Learning for Unsupervised Medical Diagnosis](https://mlsanity.com/paper/2209.10425/) (2022)
- [Keep your distance: learning dispersed embeddings on mathbbSm](https://mlsanity.com/paper/2502.08231/) (2025)
- [Maximum Mean Discrepancy with Unequal Sample Sizes via Generalized U-Statistics](https://mlsanity.com/paper/2512.13997/) (2025)
- [Are Two Datasets Close Enough With Statistical Significance? A Kernel Distributional Closeness Testing Approach](https://mlsanity.com/paper/2507.12843/) (2025)
- [Finite-Sample Unbiased Variance of MMD under Unbalanced Sampling: Exact Estimation and Quasi-Linear Computation](https://mlsanity.com/paper/2601.13874/) (2026)

## Abstract (arXiv)

The maximum mean discrepancy (MMD) is a kernel-based distance between probability distributions useful in many applications (Gretton et al. 2012), bearing a simple estimator with pleasing computational and statistical properties. Being able to efficiently estimate the variance of this estimator is very helpful to various problems in two-sample testing. Towards this end, Bounliphone et al. (2016) used the theory of U-statistics to derive estimators for the variance of an MMD estimator, and differences between two such estimators. Their estimator, however, drops lower-order terms, and is unnecessarily biased. We show in this note - extending and correcting work of Sutherland et al. (2017) - that we can find a truly unbiased estimator for the actual variance of both the squared MMD estimator and the difference of two correlated squared MMD estimators, at essentially no additional computational cost.