# Variable Selection for Kernel Two-Sample Tests

arXiv: https://arxiv.org/abs/2302.07415 · HTML: https://mlsanity.com/paper/2302.07415/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 85 references that are in the map.
- First posted on arXiv 15 Feb 2023.

## Builds on (most cited first, 29 of 29)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Distributionally Robust Stochastic Optimization with Wasserstein Distance](https://mlsanity.com/paper/1604.02199/) (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)
- [MMD Aggregated Two-Sample Test](https://mlsanity.com/paper/2110.15073/) (2021)
- [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)
- [Two-sample Testing Using Deep Learning](https://mlsanity.com/paper/1910.06239/) (2019)
- [Efficient Aggregated Kernel Tests using Incomplete U-statistics](https://mlsanity.com/paper/2206.09194/) (2022)
- [Classification Logit Two-sample Testing by Neural Networks](https://mlsanity.com/paper/1909.11298/) (2019)
- [A Witness Two-Sample Test](https://mlsanity.com/paper/2102.05573/) (2021)
- [AutoML Two-Sample Test](https://mlsanity.com/paper/2206.08843/) (2022)
- [Sparse Classification: a scalable discrete optimization perspective](https://mlsanity.com/paper/1710.01352/) (2017)
- [Two-sample Test using Projected Wasserstein Distance](https://mlsanity.com/paper/2010.11970/) (2020)
- [Grouped Variable Selection with Discrete Optimization: Computational and Statistical Perspectives](https://mlsanity.com/paper/2104.07084/) (2021)
- [Solving Large-Scale Sparse PCA to Certifiable (Near) Optimality](https://mlsanity.com/paper/2005.05195/) (2020)
- [NP-Hardness and Inapproximability of Sparse PCA](https://mlsanity.com/paper/1502.05675/) (2015)
- [Using L1-relaxation and integer programming to obtain dual bounds for sparse PCA](https://mlsanity.com/paper/1810.09062/) (2018)
- [Exact and Approximation Algorithms for Sparse PCA](https://mlsanity.com/paper/2008.12438/) (2020)
- [Solving sparse principal component analysis with global support](https://mlsanity.com/paper/2010.11152/) (2020)
- [Kernel Two-Sample Tests for Manifold Data](https://mlsanity.com/paper/2105.03425/) (2021)
- [Principal Differences Analysis: Interpretable Characterization of Differences between Distributions](https://mlsanity.com/paper/1510.08956/) (2015)
- [Beyond Symmetry: Best Submatrix Selection for the Sparse Truncated SVD](https://mlsanity.com/paper/2105.03179/) (2021)
- [Kernel Robust Hypothesis Testing](https://mlsanity.com/paper/2203.12777/) (2022)
- [A Manifold Two-Sample Test Study: Integral Probability Metric with Neural Networks](https://mlsanity.com/paper/2205.02043/) (2022)
- [On the Exactness of Dantzig-Wolfe Relaxation for Rank Constrained Optimization Problems](https://mlsanity.com/paper/2210.16191/) (2022)
- [Best Principal Submatrix Selection for the Maximum Entropy Sampling Problem: Scalable Algorithms and Performance Guarantees](https://mlsanity.com/paper/2001.08537/) (2020)
- [D-optimal Data Fusion: Exact and Approximation Algorithms](https://mlsanity.com/paper/2208.03589/) (2022)

## Built on by (most cited first, 7 of 7)

- [Variable Selection in Maximum Mean Discrepancy for Interpretable Distribution Comparison](https://mlsanity.com/paper/2311.01537/) (2023)
- [Statistical and Computational Guarantees of Kernel Max-Sliced Wasserstein Distances](https://mlsanity.com/paper/2405.15441/) (2024)
- [Post Hoc Inference for Component Attribution in Multivariate Change-Point Detection](https://mlsanity.com/paper/2607.14814/) (2026)
- [Variable Selection for Comparing High-dimensional Time-Series Data](https://mlsanity.com/paper/2412.06870/) (2024)
- [On the Limitation of Kernel Dependence Maximization for Feature Selection](https://mlsanity.com/paper/2406.06903/) (2024)
- [Word Sense Detection Leveraging Maximum Mean Discrepancy](https://mlsanity.com/paper/2506.01602/) (2025)
- [Variable Selection for Feature-Based Newsvendor](https://mlsanity.com/paper/2609.01544/) (2026)

## Abstract (arXiv)

We consider the variable selection problem for two-sample tests, aiming to select the most informative variables to determine whether two collections of samples follow the same distribution. To address this, we propose a novel framework based on the kernel maximum mean discrepancy (MMD). Our approach seeks a subset of variables with a pre-specified size that maximizes the variance-regularized kernel MMD statistic. We focus on three commonly used types of kernels: linear, quadratic, and Gaussian. From a computational perspective, we derive mixed-integer programming formulations and propose exact and approximation algorithms with performance guarantees to solve these formulations. From a statistical viewpoint, we derive the rate of testing power of our framework under appropriate conditions. These results show that the sample size requirements for the three kernels depend crucially on the number of selected variables, rather than the data dimension. Experimental results on synthetic and real datasets demonstrate the superior performance of our method, compared to other variable selection frameworks, particularly in high-dimensional settings.