# On the Optimality of Kernel-Embedding Based Goodness-of-Fit Tests

arXiv: https://arxiv.org/abs/1709.08148 · HTML: https://mlsanity.com/paper/1709.08148/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Sep 2017.

## Built on by (most cited first, 19 of 19)

- [Efficient Aggregated Kernel Tests using Incomplete U-statistics](https://mlsanity.com/paper/2206.09194/) (2022)
- [A Witness Two-Sample Test](https://mlsanity.com/paper/2102.05573/) (2021)
- [KSD Aggregated Goodness-of-fit Test](https://mlsanity.com/paper/2202.00824/) (2022)
- [Spectral Regularized Kernel Two-Sample Tests](https://mlsanity.com/paper/2212.09201/) (2022)
- [Regularized Stein Variational Gradient Flow](https://mlsanity.com/paper/2211.07861/) (2022)
- [Kernel Two-Sample Tests for Manifold Data](https://mlsanity.com/paper/2105.03425/) (2021)
- [Distance and Kernel-Based Measures for Global and Local Two-Sample Conditional Distribution Testing](https://mlsanity.com/paper/2210.08149/) (2022)
- [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)
- [Adaptive and non-adaptive minimax rates for weighted Laplacian-eigenmap based nonparametric regression](https://mlsanity.com/paper/2311.00140/) (2023)
- [Standardisation-function Kernel Stein Discrepancy: A Unifying View on Kernel Stein Discrepancy Tests for Goodness-of-fit](https://mlsanity.com/paper/2106.12105/) (2021)
- [Statistical and Geometrical properties of regularized Kernel Kullback-Leibler divergence](https://mlsanity.com/paper/2408.16543/) (2024)
- [Conditional Distributional Treatment Effects: Doubly Robust Estimation and Testing](https://mlsanity.com/paper/2603.16829/) (2026)
- [Nonasymptotic one-and two-sample tests in high dimension with unknown covariance structure](https://mlsanity.com/paper/2109.01730/) (2021)
- [Keep it Tighter -- A Story on Analytical Mean Embeddings](https://mlsanity.com/paper/2110.09516/) (2021)
- [Minimax Lower Bounds of Kernel Discrepancy Estimation: MMD, HSIC, KSD](https://mlsanity.com/paper/2607.24235/) (2026)
- [Inference for Similarity and Alignability between Noisy High-Dimensional Datasets](https://mlsanity.com/paper/2511.21074/) (2025)
- [Shrinkage Estimation of Higher Order Bochner Integrals](https://mlsanity.com/paper/2207.06357/) (2022)

## Abstract (arXiv)

The reproducing kernel Hilbert space (RKHS) embedding of distributions offers a general and flexible framework for testing problems in arbitrary domains and has attracted considerable amount of attention in recent years. To gain insights into their operating characteristics, we study here the statistical performance of such approaches within a minimax framework. Focusing on the case of goodness-of-fit tests, our analyses show that a vanilla version of the kernel-embedding based test could be suboptimal, and suggest a simple remedy by moderating the embedding. We prove that the moderated approach provides optimal tests for a wide range of deviations from the null and can also be made adaptive over a large collection of interpolation spaces. Numerical experiments are presented to further demonstrate the merits of our approach.