# A Unified View of Optimal Kernel Hypothesis Testing

arXiv: https://arxiv.org/abs/2503.07084 · HTML: https://mlsanity.com/paper/2503.07084/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 49 references that are in the map.
- First posted on arXiv 10 Mar 2025.

## Builds on (most cited first, 23 of 23)

- [RAPPOR: Randomized Aggregatable Privacy-Preserving Ordinal Response](https://mlsanity.com/paper/1407.6981/) (2014)
- [A Kernelized Stein Discrepancy for Goodness-of-fit Tests and Model Evaluation](https://mlsanity.com/paper/1602.03253/) (2016)
- [Collecting Telemetry Data Privately](https://mlsanity.com/paper/1712.01524/) (2017)
- [A Kernel Test of Goodness of Fit](https://mlsanity.com/paper/1602.02964/) (2016)
- [Measuring Sample Quality with Kernels](https://mlsanity.com/paper/1703.01717/) (2017)
- [MMD Aggregated Two-Sample Test](https://mlsanity.com/paper/2110.15073/) (2021)
- [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)
- [Blindness of score-based methods to isolated components and mixing proportions](https://mlsanity.com/paper/2008.10087/) (2020)
- [MMD-FUSE: Learning and Combining Kernels for Two-Sample Testing Without Data Splitting](https://mlsanity.com/paper/2306.08777/) (2023)
- [Large-Scale Kernel Methods for Independence Testing](https://mlsanity.com/paper/1606.07892/) (2016)
- [A Wild Bootstrap for Degenerate Kernel Tests](https://mlsanity.com/paper/1408.5404/) (2014)
- [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)
- [FastMMD: Ensemble of Circular Discrepancy for Efficient Two-Sample Test](https://mlsanity.com/paper/1405.2664/) (2014)
- [Targeted Separation and Convergence with Kernel Discrepancies](https://mlsanity.com/paper/2209.12835/) (2022)
- [Practical Kernel Tests of Conditional Independence](https://mlsanity.com/paper/2402.13196/) (2024)
- [Compress Then Test: Powerful Kernel Testing in Near-linear Time](https://mlsanity.com/paper/2301.05974/) (2023)
- [Towards Healing the Blindness of Score Matching](https://mlsanity.com/paper/2209.07396/) (2022)
- [A Practical Introduction to Kernel Discrepancies: MMD, HSIC & KSD](https://mlsanity.com/paper/2503.04820/) (2025)
- [Differentially Private Permutation Tests](https://mlsanity.com/paper/2310.19043/) (2023)
- [Robust Kernel Hypothesis Testing under Data Corruption](https://mlsanity.com/paper/2405.19912/) (2024)
- [Goodness of fit tests for high-dimensional linear models](https://mlsanity.com/paper/1511.03334/) (2015)

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

- [DUAL: Learning Diverse Kernels for Aggregated Two-sample and Independence Testing](https://mlsanity.com/paper/2510.11140/) (2025)
- [Amortized Linear-time Exact Shapley Value for Product-Kernel Methods](https://mlsanity.com/paper/2505.16516/) (2025)
- [Regularized f-Divergence Kernel Tests](https://mlsanity.com/paper/2601.19755/) (2026)
- [Kernel Quantile Embeddings and Associated Probability Metrics](https://mlsanity.com/paper/2505.20433/) (2025)
- [Aggregation of Statistical Evidence under Exchangeability](https://mlsanity.com/paper/2607.15823/) (2026)

## Abstract (arXiv)

This paper provides a unifying view of optimal kernel hypothesis testing across the MMD two-sample, HSIC independence, and KSD goodness-of-fit frameworks. Minimax optimal separation rates in the kernel and L2 metrics are presented, with two adaptive kernel selection methods (kernel pooling and aggregation), and under various testing constraints: computational efficiency, differential privacy, and robustness to data corruption. Intuition behind the derivation of the power results is provided in a unified way across the three frameworks, and open problems are highlighted.