# Spectral Regularized Kernel Goodness-of-Fit Tests

arXiv: https://arxiv.org/abs/2308.04561 · HTML: https://mlsanity.com/paper/2308.04561/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 46 references that are in the map.
- First posted on arXiv 8 Aug 2023.

## Builds on (most cited first, 11 of 11)

- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [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)
- [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)
- [Spectral Regularized Kernel Two-Sample Tests](https://mlsanity.com/paper/2212.09201/) (2022)
- [On the Optimality of Kernel-Embedding Based Goodness-of-Fit Tests](https://mlsanity.com/paper/1709.08148/) (2017)
- [Approximate Kernel PCA Using Random Features: Computational vs. Statistical Trade-off](https://mlsanity.com/paper/1706.06296/) (2017)
- [fasano.franceschini.test: An Implementation of a Multidimensional KS Test in R](https://mlsanity.com/paper/2106.10539/) (2021)

## Built on by (most cited first, 6 of 6)

- [On the Robustness of Kernel Goodness-of-Fit Tests](https://mlsanity.com/paper/2408.05854/) (2024)
- [Sobolev Regularized MMD Gradient Flow](https://mlsanity.com/paper/2605.11884/) (2026)
- [Nystr"om Kernel Stein Discrepancy](https://mlsanity.com/paper/2406.08401/) (2024)
- [Fixed-Gaussian Spectral Algorithms: Minimax Optimal Rates for Misspecified Learning and Transfer](https://mlsanity.com/paper/2501.10870/) (2025)
- [Optimal Online Change Detection via Random Fourier Features](https://mlsanity.com/paper/2505.17789/) (2025)
- [Kernel Trace Distance: Quantum Statistical Metric between Measures through RKHS Density Operators](https://mlsanity.com/paper/2507.06055/) (2025)

## Abstract (arXiv)

Maximum mean discrepancy (MMD) has enjoyed a lot of success in many machine learning and statistical applications, including non-parametric hypothesis testing, because of its ability to handle non-Euclidean data. Recently, it has been demonstrated in Balasubramanian et al.(2021) that the goodness-of-fit test based on MMD is not minimax optimal while a Tikhonov regularized version of it is, for an appropriate choice of the regularization parameter. However, the results in Balasubramanian et al. (2021) are obtained under the restrictive assumptions of the mean element being zero, and the uniform boundedness condition on the eigenfunctions of the integral operator. Moreover, the test proposed in Balasubramanian et al. (2021) is not practical as it is not computable for many kernels. In this paper, we address these shortcomings and extend the results to general spectral regularizers that include Tikhonov regularization.