# On the Optimality of Gaussian Kernel Based Nonparametric Tests against Smooth Alternatives

arXiv: https://arxiv.org/abs/1909.03302 · HTML: https://mlsanity.com/paper/1909.03302/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Sep 2019.

## Built on by (most cited first, 21 of 21)

- [MMD Aggregated Two-Sample Test](https://mlsanity.com/paper/2110.15073/) (2021)
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- [A Permutation-free Kernel Two-Sample Test](https://mlsanity.com/paper/2211.14908/) (2022)
- [Spectral Regularized Kernel Two-Sample Tests](https://mlsanity.com/paper/2212.09201/) (2022)
- [Boosting the Power of Kernel Two-Sample Tests](https://mlsanity.com/paper/2302.10687/) (2023)
- [Kernel Two-Sample Tests for Manifold Data](https://mlsanity.com/paper/2105.03425/) (2021)
- [Smoothness Adaptive Hypothesis Transfer Learning](https://mlsanity.com/paper/2402.14966/) (2024)
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- [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)
- [Self-Supervised Metric Learning in Multi-View Data: A Downstream Task Perspective](https://mlsanity.com/paper/2106.07138/) (2021)
- [Kernel-Based Tests for Likelihood-Free Hypothesis Testing](https://mlsanity.com/paper/2308.09043/) (2023)
- [Binary Hypothesis Testing for Softmax Models and Leverage Score Models](https://mlsanity.com/paper/2405.06003/) (2024)
- [ReLU integral probability metric and its applications](https://mlsanity.com/paper/2504.18897/) (2025)
- [Discussion of `Multiscale Fisher's Independence Test for Multivariate Dependence'](https://mlsanity.com/paper/2206.11142/) (2022)
- [A Two-Sample Test of Text Generation Similarity](https://mlsanity.com/paper/2505.05269/) (2025)
- [A nonparametric two-sample test using a parametric integral probability metric](https://mlsanity.com/paper/2606.16941/) (2026)

## Abstract (arXiv)

Nonparametric tests via kernel embedding of distributions have witnessed a great deal of practical successes in recent years. However, statistical properties of these tests are largely unknown beyond consistency against a fixed alternative. To fill in this void, we study here the asymptotic properties of goodness-of-fit, homogeneity and independence tests using Gaussian kernels, arguably the most popular and successful among such tests. Our results provide theoretical justifications for this common practice by showing that tests using Gaussian kernel with an appropriately chosen scaling parameter are minimax optimal against smooth alternatives in all three settings. In addition, our analysis also pinpoints the importance of choosing a diverging scaling parameter when using Gaussian kernels and suggests a data-driven choice of the scaling parameter that yields tests optimal, up to an iterated logarithmic factor, over a wide range of smooth alternatives. Numerical experiments are also presented to further demonstrate the practical merits of the methodology.