# Boosting the Power of Kernel Two-Sample Tests

arXiv: https://arxiv.org/abs/2302.10687 · HTML: https://mlsanity.com/paper/2302.10687/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 68 references that are in the map.
- First posted on arXiv 21 Feb 2023.

## Builds on (most cited first, 14 of 14)

- [Revisiting Classifier Two-Sample Tests](https://mlsanity.com/paper/1610.06545/) (2016)
- [On Wasserstein Two Sample Testing and Related Families of Nonparametric Tests](https://mlsanity.com/paper/1509.02237/) (2015)
- [Learning Deep Kernels for Non-Parametric Two-Sample Tests](https://mlsanity.com/paper/2002.09116/) (2020)
- [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 new graph-based two-sample test for multivariate and object data](https://mlsanity.com/paper/1307.6294/) (2013)
- [Multivariate Rank-based Distribution-free Nonparametric Testing using Measure Transportation](https://mlsanity.com/paper/1909.08733/) (2019)
- [On the Optimality of Gaussian Kernel Based Nonparametric Tests against Smooth Alternatives](https://mlsanity.com/paper/1909.03302/) (2019)
- [A Permutation-free Kernel Two-Sample Test](https://mlsanity.com/paper/2211.14908/) (2022)
- [Learning Kernel Tests Without Data Splitting](https://mlsanity.com/paper/2006.02286/) (2020)
- [Learning Sparse Feature Representations using Probabilistic Quadtrees and Deep Belief Nets](https://mlsanity.com/paper/1509.03413/) (2015)
- [Generalized Kernel Two-Sample Tests](https://mlsanity.com/paper/2011.06127/) (2020)
- [On high-dimensional modifications of some graph-based two-sample tests](https://mlsanity.com/paper/1806.02138/) (2018)

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

- [MMD-FUSE: Learning and Combining Kernels for Two-Sample Testing Without Data Splitting](https://mlsanity.com/paper/2306.08777/) (2023)
- [Robust Kernel Hypothesis Testing under Data Corruption](https://mlsanity.com/paper/2405.19912/) (2024)
- [A Kernel-Based Conditional Two-Sample Test Using Nearest Neighbors (with Applications to Calibration, Regression Curves, and Simulation-Based Inference)](https://mlsanity.com/paper/2407.16550/) (2024)
- [DUAL: Learning Diverse Kernels for Aggregated Two-sample and Independence Testing](https://mlsanity.com/paper/2510.11140/) (2025)
- [Kernel Embeddings and the Separation of Measure Phenomenon](https://mlsanity.com/paper/2505.04613/) (2025)
- [Likelihood Ratio Tests by Kernel Gaussian Embedding](https://mlsanity.com/paper/2508.07982/) (2025)
- [Kernel Two-Sample Testing via Directional Components Analysis](https://mlsanity.com/paper/2508.08564/) (2025)
- [Kernel-based Equalized Odds: A Quantification of Accuracy-Fairness Trade-off in Fair Representation Learning](https://mlsanity.com/paper/2508.15084/) (2025)
- [One-shot Conditional Sampling: MMD meets Nearest Neighbors](https://mlsanity.com/paper/2509.25507/) (2025)
- [USAD: Uncertainty-aware Statistical Adversarial Detection](https://mlsanity.com/paper/2606.27832/) (2026)

## Abstract (arXiv)

The kernel two-sample test based on the maximum mean discrepancy (MMD) is one of the most popular methods for detecting differences between two distributions over general metric spaces. In this paper we propose a method to boost the power of the kernel test by combining MMD estimates over multiple kernels using their Mahalanobis distance. We derive the asymptotic null distribution of the proposed test statistic and use a multiplier bootstrap approach to efficiently compute the rejection region. The resulting test is universally consistent and, since it is obtained by aggregating over a collection of kernels/bandwidths, is more powerful in detecting a wide range of alternatives in finite samples. We also derive the distribution of the test statistic for both fixed and local contiguous alternatives. The latter, in particular, implies that the proposed test is statistically efficient, that is, it has non-trivial asymptotic (Pitman) efficiency. The consistency properties of the Mahalanobis and other natural aggregation methods are also explored when the number of kernels is allowed to grow with the sample size. Extensive numerical experiments are performed on both synthetic and real-world datasets to illustrate the efficacy of the proposed method over single kernel tests. The computational complexity of the proposed method is also studied, both theoretically and in simulations. Our asymptotic results rely on deriving the joint distribution of MMD estimates using the framework of multiple stochastic integrals, which is more broadly useful, specifically, in understanding the efficiency properties of recently proposed adaptive MMD tests based on kernel aggregation and also in developing more computationally efficient (linear time) tests that combine multiple kernels. We conclude with an application of the Mahalanobis aggregation method for kernels with diverging scaling parameters.