# MMD-FUSE: Learning and Combining Kernels for Two-Sample Testing Without Data Splitting

arXiv: https://arxiv.org/abs/2306.08777 · HTML: https://mlsanity.com/paper/2306.08777/

## Key facts

- Cited by 33 papers in the map (data through 14 Oct 2026).
- Builds on 58 of its 107 references that are in the map.
- First posted on arXiv 14 Jun 2023.

## Builds on (most cited first, 30 of 58)

- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [Momentum Contrast for Unsupervised Visual Representation Learning](https://mlsanity.com/paper/1911.05722/) (2019)
- [Bootstrap your own latent: A new approach to self-supervised Learning](https://mlsanity.com/paper/2006.07733/) (2020)
- [Exploring Simple Siamese Representation Learning](https://mlsanity.com/paper/2011.10566/) (2020)
- [Unsupervised Learning of Visual Features by Contrasting Cluster Assignments](https://mlsanity.com/paper/2006.09882/) (2020)
- [Improved Baselines with Momentum Contrastive Learning](https://mlsanity.com/paper/2003.04297/) (2020)
- [Barlow Twins: Self-Supervised Learning via Redundancy Reduction](https://mlsanity.com/paper/2103.03230/) (2021)
- [Do ImageNet Classifiers Generalize to ImageNet?](https://mlsanity.com/paper/1902.10811/) (2019)
- [Big Self-Supervised Models are Strong Semi-Supervised Learners](https://mlsanity.com/paper/2006.10029/) (2020)
- [Computing Nonvacuous Generalization Bounds for Deep (Stochastic) Neural Networks with Many More Parameters than Training Data](https://mlsanity.com/paper/1703.11008/) (2017)
- [Revisiting Classifier Two-Sample Tests](https://mlsanity.com/paper/1610.06545/) (2016)
- [User-friendly introduction to PAC-Bayes bounds](https://mlsanity.com/paper/2110.11216/) (2021)
- [A Kernelized Stein Discrepancy for Goodness-of-fit Tests and Model Evaluation](https://mlsanity.com/paper/1602.03253/) (2016)
- [A Kernel Test of Goodness of Fit](https://mlsanity.com/paper/1602.02964/) (2016)
- [A Primer on PAC-Bayesian Learning](https://mlsanity.com/paper/1901.05353/) (2019)
- [Learning Deep Kernels for Non-Parametric Two-Sample Tests](https://mlsanity.com/paper/2002.09116/) (2020)
- [Non-Vacuous Generalization Bounds at the ImageNet Scale: A PAC-Bayesian Compression Approach](https://mlsanity.com/paper/1804.05862/) (2018)
- [Tighter risk certificates for neural networks](https://mlsanity.com/paper/2007.12911/) (2020)
- [Generative Models and Model Criticism via Optimized Maximum Mean Discrepancy](https://mlsanity.com/paper/1611.04488/) (2016)
- [Entropy-SGD optimizes the prior of a PAC-Bayes bound: Generalization properties of Entropy-SGD and data-dependent priors](https://mlsanity.com/paper/1712.09376/) (2017)
- [Self-Supervised Learning with Kernel Dependence Maximization](https://mlsanity.com/paper/2106.08320/) (2021)
- [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)
- [Kernel Thinning](https://mlsanity.com/paper/2105.05842/) (2021)
- [Galaxy Zoo DECaLS: Detailed Visual Morphology Measurements from Volunteers and Deep Learning for 314,000 Galaxies](https://mlsanity.com/paper/2102.08414/) (2021)
- [Second Order PAC-Bayesian Bounds for the Weighted Majority Vote](https://mlsanity.com/paper/2007.13532/) (2020)
- [A Wild Bootstrap for Degenerate Kernel Tests](https://mlsanity.com/paper/1408.5404/) (2014)
- [Differentiable PAC-Bayes Objectives with Partially Aggregated Neural Networks](https://mlsanity.com/paper/2006.12228/) (2020)
- [Two-sample Testing Using Deep Learning](https://mlsanity.com/paper/1910.06239/) (2019)

## Built on by (most cited first, 30 of 33)

- [Practical Kernel Tests of Conditional Independence](https://mlsanity.com/paper/2402.13196/) (2024)
- [A Practical Guide to Sample-based Statistical Distances for Evaluating Generative Models in Science](https://mlsanity.com/paper/2403.12636/) (2024)
- [A Practical Introduction to Kernel Discrepancies: MMD, HSIC & KSD](https://mlsanity.com/paper/2503.04820/) (2025)
- [Neural-Kernel Conditional Mean Embeddings](https://mlsanity.com/paper/2403.10859/) (2024)
- [Robust Kernel Hypothesis Testing under Data Corruption](https://mlsanity.com/paper/2405.19912/) (2024)
- [Particle Semi-Implicit Variational Inference](https://mlsanity.com/paper/2407.00649/) (2024)
- [Multiple testing for signal-agnostic searches of new physics with machine learning](https://mlsanity.com/paper/2408.12296/) (2024)
- [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)
- [DUAL: Learning Diverse Kernels for Aggregated Two-sample and Independence Testing](https://mlsanity.com/paper/2510.11140/) (2025)
- [A Unified View of Optimal Kernel Hypothesis Testing](https://mlsanity.com/paper/2503.07084/) (2025)
- [Knowledge Distillation Detection for Open-weights Models](https://mlsanity.com/paper/2510.02302/) (2025)
- [Model-Agnostic Signal Discovery with Machine Learning: Bridging the Gap Between Theory and Practice](https://mlsanity.com/paper/2605.31103/) (2026)
- [Zero-Flow Two-Sample Tests](https://mlsanity.com/paper/2607.21542/) (2026)
- [Learning Representations for Independence Testing](https://mlsanity.com/paper/2409.06890/) (2024)
- [AutoSciDACT: Automated Scientific Discovery through Contrastive Embedding and Hypothesis Testing](https://mlsanity.com/paper/2510.21935/) (2025)
- [Regularized f-Divergence Kernel Tests](https://mlsanity.com/paper/2601.19755/) (2026)
- [KAIROS: Scalable Model-Agnostic Data Valuation](https://mlsanity.com/paper/2506.23799/) (2025)
- [Conformal C2ST: Turning weak classifiers into strong two-sample tests](https://mlsanity.com/paper/2507.17026/) (2025)
- [Anchor-based Maximum Discrepancy for Relative Similarity Testing](https://mlsanity.com/paper/2510.10477/) (2025)
- [Minimax-Optimal Two-Sample Test with Sliced Wasserstein](https://mlsanity.com/paper/2510.27498/) (2025)
- [Sparse, self-organizing ensembles of local kernels detect rare statistical anomalies](https://mlsanity.com/paper/2511.03095/) (2025)
- [LOTTERY: Learning from Reference-Only Samples in Two-Sample Testing under Size Asymmetry](https://mlsanity.com/paper/2606.08460/) (2026)
- [A Gradient Flow Perspective on Minimum MMD Estimation](https://mlsanity.com/paper/2607.03871/) (2026)
- [Kernel Quantile Embeddings and Associated Probability Metrics](https://mlsanity.com/paper/2505.20433/) (2025)
- [Weakly Supervised Object Segmentation by Background Conditional Divergence](https://mlsanity.com/paper/2506.22505/) (2025)
- [Are Two Datasets Close Enough With Statistical Significance? A Kernel Distributional Closeness Testing Approach](https://mlsanity.com/paper/2507.12843/) (2025)
- [Classical and quantum kernel fusion for two-sample testing](https://mlsanity.com/paper/2511.20941/) (2025)
- [Learning to Test: Latent Representation Learning for Dynamic Instability Detection](https://mlsanity.com/paper/2604.10967/) (2026)
- [Kernel Selection is Model Selection: A Unified Complexity-Penalized Approach for MMD Two-Sample Tests](https://mlsanity.com/paper/2605.06883/) (2026)

## Abstract (arXiv)

We propose novel statistics which maximise the power of a two-sample test based on the Maximum Mean Discrepancy (MMD), by adapting over the set of kernels used in defining it. For finite sets, this reduces to combining (normalised) MMD values under each of these kernels via a weighted soft maximum. Exponential concentration bounds are proved for our proposed statistics under the null and alternative. We further show how these kernels can be chosen in a data-dependent but permutation-independent way, in a well-calibrated test, avoiding data splitting. This technique applies more broadly to general permutation-based MMD testing, and includes the use of deep kernels with features learnt using unsupervised models such as auto-encoders. We highlight the applicability of our MMD-FUSE test on both synthetic low-dimensional and real-world high-dimensional data, and compare its performance in terms of power against current state-of-the-art kernel tests.