# Minimax Estimation of Kernel Stein Discrepancy: Trace versus Hilbert-Schmidt Scales

arXiv: https://arxiv.org/abs/2607.03367 · HTML: https://mlsanity.com/paper/2607.03367/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 26 references that are in the map.
- First posted on arXiv 3 Jul 2026.

## Builds on (most cited first, 17 of 17)

- [Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm](https://mlsanity.com/paper/1608.04471/) (2016)
- [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)
- [Measuring Sample Quality with Kernels](https://mlsanity.com/paper/1703.01717/) (2017)
- [Measuring Sample Quality with Stein's Method](https://mlsanity.com/paper/1506.03039/) (2015)
- [Minimum Stein Discrepancy Estimators](https://mlsanity.com/paper/1906.08283/) (2019)
- [Stein Points](https://mlsanity.com/paper/1803.10161/) (2018)
- [A Linear-Time Kernel Goodness-of-Fit Test](https://mlsanity.com/paper/1705.07673/) (2017)
- [Random Feature Stein Discrepancies](https://mlsanity.com/paper/1806.07788/) (2018)
- [A Kernel Stein Test for Comparing Latent Variable Models](https://mlsanity.com/paper/1907.00586/) (2019)
- [Improved Finite-Particle Convergence Rates for Stein Variational Gradient Descent](https://mlsanity.com/paper/2409.08469/) (2024)
- [Minimax Optimal Goodness-of-Fit Testing with Kernel Stein Discrepancy](https://mlsanity.com/paper/2404.08278/) (2024)
- [Probabilistic Inference and Learning with Stein's Method](https://mlsanity.com/paper/2603.07467/) (2026)
- [Nystr"om Kernel Stein Discrepancy](https://mlsanity.com/paper/2406.08401/) (2024)
- [The Minimax Lower Bound of Kernel Stein Discrepancy Estimation](https://mlsanity.com/paper/2510.15058/) (2025)
- [Nystr"om Kernel Stein Discrepancy Tests](https://mlsanity.com/paper/2605.25173/) (2026)

## Built on by (most cited first, 3 of 3)

- [Minimax Lower Bounds of Kernel Discrepancy Estimation: MMD, HSIC, KSD](https://mlsanity.com/paper/2607.24235/) (2026)
- [Unbounded Characteristic and Universal Kernels](https://mlsanity.com/paper/2610.09731/) (2026)
- [Conditional Kernel Stein Discrepancy](https://mlsanity.com/paper/2610.11863/) (2026)

## Abstract (arXiv)

Kernel Stein Discrepancy (KSD) compares a sample to a fixed target distribution known only through its score, and is widely used for goodness-of-fit testing, sample quality assessment, and approximate inference. We study the estimation of operatornameKSD(P0,P) from n independent observations and identify the sharp spectral constant governing the minimax risk: it is the Hilbert-Schmidt norm of the Stein covariance operator C_star, giving the minimax scale √|C_star|HS/n. This scale is attained by the positive-part square-root U-statistic, whereas the standard plug-in V-statistic remains at the trace scale √operatornametr(C_star)/n and is therefore suboptimal by the fourth root of the effective rank of C_star; for a Gaussian target with a fixed-bandwidth Gaussian kernel this factor is exponential in the dimension.