# Counterfactual Explanations for Deep Two-Sample Testing

arXiv: https://arxiv.org/abs/2606.04009 · HTML: https://mlsanity.com/paper/2606.04009/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 26 references that are in the map.
- First posted on arXiv 29 May 2026.

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

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [The Unreasonable Effectiveness of Deep Features as a Perceptual Metric](https://mlsanity.com/paper/1801.03924/) (2018)
- [Grad-CAM: Visual Explanations from Deep Networks via Gradient-based Localization](https://mlsanity.com/paper/1610.02391/) (2016)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Axiomatic Attribution for Deep Networks](https://mlsanity.com/paper/1703.01365/) (2017)
- [Learning Important Features Through Propagating Activation Differences](https://mlsanity.com/paper/1704.02685/) (2017)
- [Striving for Simplicity: The All Convolutional Net](https://mlsanity.com/paper/1412.6806/) (2014)
- [Diffusion Autoencoders: Toward a Meaningful and Decodable Representation](https://mlsanity.com/paper/2111.15640/) (2021)
- [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)
- [Two-sample Testing Using Deep Learning](https://mlsanity.com/paper/1910.06239/) (2019)
- [Global Counterfactual Directions](https://mlsanity.com/paper/2404.12488/) (2024)
- [Benchmarks, Algorithms, and Metrics for Hierarchical Disentanglement](https://mlsanity.com/paper/2102.05185/) (2021)
- [A Unified Data Representation Learning for Non-parametric Two-sample Testing](https://mlsanity.com/paper/2412.00613/) (2024)

## Abstract (arXiv)

Two-sample testing is a fundamental tool for detecting distributional differences across scientific domains, but classical tests (including kernel-based tests) can be ineffective on high-dimensional structured data such as images. Recent deep two-sample tests improve sensitivity in these settings by learning informative representations, yet they provide limited insight into which data features drive rejection of the null hypothesis H0. To address this issue, we propose a counterfactual explanation framework for deep two-sample testing that generates sample-level edits moving observations from a source group toward a target group while explicitly reducing the discrepancy measured by the test. Our method combines a diffusion autoencoder with a pretrained deep two-sample test model and optimizes a maximum mean discrepancy (MMD) objective in the test model's representation space to produce plausible counterfactuals. We quantify distribution-level effects through changes in the test statistic and the resulting two-sample p-values. We evaluate the method on synthetic 2D shape datasets and two MRI cohorts. Across both settings, the counterfactual transformations consistently increase p-values relative to the original samples, indicating that the edited source set becomes statistically closer to the target distribution under the test. We measure minimality using LPIPS to ensure the counterfactuals remain close to the original samples. The resulting edits provide interpretable evidence of the features associated with the detected group differences. On MRI, the localized changes are consistent with known anatomical differences between cohorts.