# Spectral Regularized Kernel Two-Sample Tests

arXiv: https://arxiv.org/abs/2212.09201 · HTML: https://mlsanity.com/paper/2212.09201/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 50 references that are in the map.
- First posted on arXiv 19 Dec 2022.

## Builds on (most cited first, 8 of 8)

- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [MMD Aggregated Two-Sample Test](https://mlsanity.com/paper/2110.15073/) (2021)
- [Kernel Distribution Embeddings: Universal Kernels, Characteristic Kernels and Kernel Metrics on Distributions](https://mlsanity.com/paper/1604.05251/) (2016)
- [Randomized sketches for kernels: Fast and optimal non-parametric regression](https://mlsanity.com/paper/1501.06195/) (2015)
- [On the Optimality of Gaussian Kernel Based Nonparametric Tests against Smooth Alternatives](https://mlsanity.com/paper/1909.03302/) (2019)
- [On the Optimality of Kernel-Embedding Based Goodness-of-Fit Tests](https://mlsanity.com/paper/1709.08148/) (2017)
- [Statistical Optimality and Computational Efficiency of Nystr"om Kernel PCA](https://mlsanity.com/paper/2105.08875/) (2021)
- [Approximate Kernel PCA Using Random Features: Computational vs. Statistical Trade-off](https://mlsanity.com/paper/1706.06296/) (2017)

## Built on by (most cited first, 20 of 20)

- [MMD-FUSE: Learning and Combining Kernels for Two-Sample Testing Without Data Splitting](https://mlsanity.com/paper/2306.08777/) (2023)
- [Robust Kernel Hypothesis Testing under Data Corruption](https://mlsanity.com/paper/2405.19912/) (2024)
- [On the Robustness of Kernel Goodness-of-Fit Tests](https://mlsanity.com/paper/2408.05854/) (2024)
- [Spectral Regularized Kernel Goodness-of-Fit Tests](https://mlsanity.com/paper/2308.04561/) (2023)
- [A Scalable Nystrom-Based Kernel Two-Sample Test with Permutations](https://mlsanity.com/paper/2502.13570/) (2025)
- [Sobolev Regularized MMD Gradient Flow](https://mlsanity.com/paper/2605.11884/) (2026)
- [Kernelized Cumulants: Beyond Kernel Mean Embeddings](https://mlsanity.com/paper/2301.12466/) (2023)
- [Nystr"om M-Hilbert-Schmidt Independence Criterion](https://mlsanity.com/paper/2302.09930/) (2023)
- [Nystr"om Kernel Stein Discrepancy](https://mlsanity.com/paper/2406.08401/) (2024)
- [Statistical and Geometrical properties of regularized Kernel Kullback-Leibler divergence](https://mlsanity.com/paper/2408.16543/) (2024)
- [Kernel-Based Testing for Single-Cell Differential Analysis](https://mlsanity.com/paper/2307.08509/) (2023)
- [Binary Losses for Density Ratio Estimation](https://mlsanity.com/paper/2407.01371/) (2024)
- [Fixed-Gaussian Spectral Algorithms: Minimax Optimal Rates for Misspecified Learning and Transfer](https://mlsanity.com/paper/2501.10870/) (2025)
- [Kernel Embeddings and the Separation of Measure Phenomenon](https://mlsanity.com/paper/2505.04613/) (2025)
- [Optimal Online Change Detection via Random Fourier Features](https://mlsanity.com/paper/2505.17789/) (2025)
- [Likelihood Ratio Tests by Kernel Gaussian Embedding](https://mlsanity.com/paper/2508.07982/) (2025)
- [Beyond Cross-Validation: Adaptive Parameter Selection for Kernel-Based Gradient Descents](https://mlsanity.com/paper/2603.03401/) (2026)
- [Are Two Datasets Close Enough With Statistical Significance? A Kernel Distributional Closeness Testing Approach](https://mlsanity.com/paper/2507.12843/) (2025)
- [Perturbative methods for non-parametric instrumental variable](https://mlsanity.com/paper/2606.00322/) (2026)
- [Markov Chain Convergence in Kernel-Induced Geometry](https://mlsanity.com/paper/2610.13916/) (2026)

## Abstract (arXiv)

Over the last decade, an approach that has gained a lot of popularity to tackle nonparametric testing problems on general (i.e., non-Euclidean) domains is based on the notion of reproducing kernel Hilbert space (RKHS) embedding of probability distributions. The main goal of our work is to understand the optimality of two-sample tests constructed based on this approach. First, we show the popular MMD (maximum mean discrepancy) two-sample test to be not optimal in terms of the separation boundary measured in Hellinger distance. Second, we propose a modification to the MMD test based on spectral regularization by taking into account the covariance information (which is not captured by the MMD test) and prove the proposed test to be minimax optimal with a smaller separation boundary than that achieved by the MMD test. Third, we propose an adaptive version of the above test which involves a data-driven strategy to choose the regularization parameter and show the adaptive test to be almost minimax optimal up to a logarithmic factor. Moreover, our results hold for the permutation variant of the test where the test threshold is chosen elegantly through the permutation of the samples. Through numerical experiments on synthetic and real data, we demonstrate the superior performance of the proposed test in comparison to the MMD test and other popular tests in the literature.