# Variable Selection in Maximum Mean Discrepancy for Interpretable Distribution Comparison

arXiv: https://arxiv.org/abs/2311.01537 · HTML: https://mlsanity.com/paper/2311.01537/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 61 references that are in the map.
- First posted on arXiv 2 Nov 2023.

## Builds on (most cited first, 13 of 13)

- [StarGAN v2: Diverse Image Synthesis for Multiple Domains](https://mlsanity.com/paper/1912.01865/) (2019)
- [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)
- [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)
- [Counterfactual Mean Embeddings](https://mlsanity.com/paper/1805.08845/) (2018)
- [Model Based Explanations of Concept Drift](https://mlsanity.com/paper/2303.09331/) (2023)
- [Post Selection Inference with Incomplete Maximum Mean Discrepancy Estimator](https://mlsanity.com/paper/1802.06226/) (2018)
- [Global and Local Two-Sample Tests via Regression](https://mlsanity.com/paper/1812.08927/) (2018)
- [Variable Selection for Kernel Two-Sample Tests](https://mlsanity.com/paper/2302.07415/) (2023)
- [Principal Differences Analysis: Interpretable Characterization of Differences between Distributions](https://mlsanity.com/paper/1510.08956/) (2015)
- [More Powerful Selective Kernel Tests for Feature Selection](https://mlsanity.com/paper/1910.06134/) (2019)

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

- [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)
- [MMD-Flagger: Leveraging Maximum Mean Discrepancy to Detect Hallucinations](https://mlsanity.com/paper/2506.01367/) (2025)
- [Word Sense Detection Leveraging Maximum Mean Discrepancy](https://mlsanity.com/paper/2506.01602/) (2025)

## Abstract (arXiv)

We study two-sample variable selection: identifying variables that discriminate between the distributions of two sets of data vectors. Such variables help scientists understand the mechanisms behind dataset discrepancies. Although domain-specific methods exist (e.g., in medical imaging, genetics, and computational social science), a general framework remains underdeveloped. We make two separate contributions. (i) We introduce a mathematical notion of the discriminating set of variables: the largest subset containing no variables whose marginals are identical across the two distributions and independent of the remaining variables. We prove this set is uniquely defined and establish further properties, making it a suitable ground truth for theory and evaluation. (ii) We propose two methods for two-sample variable selection that assign weights to variables and optimise them to maximise the power of a kernel two-sample test while enforcing sparsity to downweight redundant variables. To select the regularisation parameter - unknown in practice, as it controls the number of selected variables - we develop two data-driven procedures to balance recall and precision. Synthetic experiments show improved performance over baselines, and we illustrate the approach on two applications using datasets from water-pipe and traffic networks.