# A Witness Two-Sample Test

arXiv: https://arxiv.org/abs/2102.05573 · HTML: https://mlsanity.com/paper/2102.05573/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 41 references that are in the map.
- First posted on arXiv 10 Feb 2021.

## Builds on (most cited first, 15 of 15)

- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Revisiting Classifier Two-Sample Tests](https://mlsanity.com/paper/1610.06545/) (2016)
- [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)
- [Kernel methods through the roof: handling billions of points efficiently](https://mlsanity.com/paper/2006.10350/) (2020)
- [FALKON: An Optimal Large Scale Kernel Method](https://mlsanity.com/paper/1705.10958/) (2017)
- [Interpretable Distribution Features with Maximum Testing Power](https://mlsanity.com/paper/1605.06796/) (2016)
- [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)
- [Two-sample Testing Using Deep Learning](https://mlsanity.com/paper/1910.06239/) (2019)
- [Classification Logit Two-sample Testing by Neural Networks](https://mlsanity.com/paper/1909.11298/) (2019)
- [A new graph-based two-sample test for multivariate and object data](https://mlsanity.com/paper/1307.6294/) (2013)
- [On the Optimality of Gaussian Kernel Based Nonparametric Tests against Smooth Alternatives](https://mlsanity.com/paper/1909.03302/) (2019)
- [On the Optimality of Kernel-Embedding Based Goodness-of-Fit Tests](https://mlsanity.com/paper/1709.08148/) (2017)
- [Learning Kernel Tests Without Data Splitting](https://mlsanity.com/paper/2006.02286/) (2020)

## Built on by (most cited first, 23 of 23)

- [MMD Aggregated Two-Sample Test](https://mlsanity.com/paper/2110.15073/) (2021)
- [MMD-FUSE: Learning and Combining Kernels for Two-Sample Testing Without Data Splitting](https://mlsanity.com/paper/2306.08777/) (2023)
- [Efficient Aggregated Kernel Tests using Incomplete U-statistics](https://mlsanity.com/paper/2206.09194/) (2022)
- [KSD Aggregated Goodness-of-fit Test](https://mlsanity.com/paper/2202.00824/) (2022)
- [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)
- [A Practical Guide to Sample-based Statistical Distances for Evaluating Generative Models in Science](https://mlsanity.com/paper/2403.12636/) (2024)
- [Auditing and Generating Synthetic Data with Controllable Trust Trade-offs](https://mlsanity.com/paper/2304.10819/) (2023)
- [Variable Selection for Kernel Two-Sample Tests](https://mlsanity.com/paper/2302.07415/) (2023)
- [A Unified Data Representation Learning for Non-parametric Two-sample Testing](https://mlsanity.com/paper/2412.00613/) (2024)
- [A Scalable Nystrom-Based Kernel Two-Sample Test with Permutations](https://mlsanity.com/paper/2502.13570/) (2025)
- [Signature Maximum Mean Discrepancy Two-Sample Statistical Tests](https://mlsanity.com/paper/2506.01718/) (2025)
- [Conditional Distributional Treatment Effects: Doubly Robust Estimation and Testing](https://mlsanity.com/paper/2603.16829/) (2026)
- [Zero-Flow Two-Sample Tests](https://mlsanity.com/paper/2607.21542/) (2026)
- [Kernel-Based Testing for Single-Cell Differential Analysis](https://mlsanity.com/paper/2307.08509/) (2023)
- [Learning Representations for Independence Testing](https://mlsanity.com/paper/2409.06890/) (2024)
- [Sparse, self-organizing ensembles of local kernels detect rare statistical anomalies](https://mlsanity.com/paper/2511.03095/) (2025)
- [Maximum Mean Discrepancy with Unequal Sample Sizes via Generalized U-Statistics](https://mlsanity.com/paper/2512.13997/) (2025)
- [LOTTERY: Learning from Reference-Only Samples in Two-Sample Testing under Size Asymmetry](https://mlsanity.com/paper/2606.08460/) (2026)
- [Two-Sample Testing via Path-based Inference](https://mlsanity.com/paper/2610.05684/) (2026)
- [On Ranking-based Tests of Independence](https://mlsanity.com/paper/2403.07464/) (2024)
- [Two-Sample Testing via Generative Processes](https://mlsanity.com/paper/2610.08277/) (2026)
- [Learning to Detect Shifts in Unstructured Data Streams via Monitoring-by-Betting](https://mlsanity.com/paper/2610.17399/) (2026)

## Abstract (arXiv)

The Maximum Mean Discrepancy (MMD) has been the state-of-the-art nonparametric test for tackling the two-sample problem. Its statistic is given by the difference in expectations of the witness function, a real-valued function defined as a weighted sum of kernel evaluations on a set of basis points. Typically the kernel is optimized on a training set, and hypothesis testing is performed on a separate test set to avoid overfitting (i.e., control type-I error). That is, the test set is used to simultaneously estimate the expectations and define the basis points, while the training set only serves to select the kernel and is discarded. In this work, we propose to use the training data to also define the weights and the basis points for better data efficiency. We show that 1) the new test is consistent and has a well-controlled type-I error; 2) the optimal witness function is given by a precision-weighted mean in the reproducing kernel Hilbert space associated with the kernel; and 3) the test power of the proposed test is comparable or exceeds that of the MMD and other modern tests, as verified empirically on challenging synthetic and real problems (e.g., Higgs data).