# Kernel Two-Sample Tests in High Dimension: Interplay Between Moment Discrepancy and Dimension-and-Sample Orders

arXiv: https://arxiv.org/abs/2201.00073 · HTML: https://mlsanity.com/paper/2201.00073/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 48 references that are in the map.
- First posted on arXiv 31 Dec 2021.

## Builds on (most cited first, 20 of 20)

- [Demystifying MMD GANs](https://mlsanity.com/paper/1801.01401/) (2018)
- [Deep Transfer Learning with Joint Adaptation Networks](https://mlsanity.com/paper/1605.06636/) (2016)
- [Generative Moment Matching Networks](https://mlsanity.com/paper/1502.02761/) (2015)
- [A Kernelized Stein Discrepancy for Goodness-of-fit Tests and Model Evaluation](https://mlsanity.com/paper/1602.03253/) (2016)
- [The Variational Fair Autoencoder](https://mlsanity.com/paper/1511.00830/) (2015)
- [Just Interpolate: Kernel "Ridgeless" Regression Can Generalize](https://mlsanity.com/paper/1808.00387/) (2018)
- [Linearized two-layers neural networks in high dimension](https://mlsanity.com/paper/1904.12191/) (2019)
- [Mind the Class Weight Bias: Weighted Maximum Mean Discrepancy for Unsupervised Domain Adaptation](https://mlsanity.com/paper/1705.00609/) (2017)
- [When Do Neural Networks Outperform Kernel Methods?](https://mlsanity.com/paper/2006.13409/) (2020)
- [A Kernel Test of Goodness of Fit](https://mlsanity.com/paper/1602.02964/) (2016)
- [Deep Knockoffs](https://mlsanity.com/paper/1811.06687/) (2018)
- [Support points](https://mlsanity.com/paper/1609.01811/) (2016)
- [A Nonparametric Approach for Multiple Change Point Analysis of Multivariate Data](https://mlsanity.com/paper/1306.4933/) (2013)
- [A new graph-based two-sample test for multivariate and object data](https://mlsanity.com/paper/1307.6294/) (2013)
- [Adaptivity and Computation-Statistics Tradeoffs for Kernel and Distance based High Dimensional Two Sample Testing](https://mlsanity.com/paper/1508.00655/) (2015)
- [A New Framework for Distance and Kernel-based Metrics in High Dimensions](https://mlsanity.com/paper/1909.13469/) (2019)
- [Distance-based and RKHS-based Dependence Metrics in High Dimension](https://mlsanity.com/paper/1902.03291/) (2019)
- [Asymptotic Distributions of High-Dimensional Distance Correlation Inference](https://mlsanity.com/paper/1910.12970/) (2019)
- [High-dimensional Change-point Detection Using Generalized Homogeneity Metrics](https://mlsanity.com/paper/2105.08976/) (2021)
- [Interpoint Distance Based Two Sample Tests in High Dimension](https://mlsanity.com/paper/1902.07279/) (2019)

## Built on by (most cited first, 5 of 5)

- [Distance and Kernel-Based Measures for Global and Local Two-Sample Conditional Distribution Testing](https://mlsanity.com/paper/2210.08149/) (2022)
- [A Unified Data Representation Learning for Non-parametric Two-sample Testing](https://mlsanity.com/paper/2412.00613/) (2024)
- [A Two-Sample Test of Text Generation Similarity](https://mlsanity.com/paper/2505.05269/) (2025)
- [Moment Expansions of the Energy Distance](https://mlsanity.com/paper/2505.20647/) (2025)
- [Machine-Learning-Assisted Comparison of Regression Functions](https://mlsanity.com/paper/2510.24714/) (2025)

## Abstract (arXiv)

Motivated by the increasing use of kernel-based metrics for high-dimensional and large-scale data, we study the asymptotic behavior of kernel two-sample tests when the dimension and sample sizes both diverge to infinity. We focus on the maximum mean discrepancy (MMD) using isotropic kernel, including MMD with the Gaussian kernel and the Laplace kernel, and the energy distance as special cases. We derive asymptotic expansions of the kernel two-sample statistics, based on which we establish the central limit theorem (CLT) under both the null hypothesis and the local and fixed alternatives. The new non-null CLT results allow us to perform asymptotic exact power analysis, which reveals a delicate interplay between the moment discrepancy that can be detected by the kernel two-sample tests and the dimension-and-sample orders. The asymptotic theory is further corroborated through numerical studies.