# KSD Aggregated Goodness-of-fit Test

arXiv: https://arxiv.org/abs/2202.00824 · HTML: https://mlsanity.com/paper/2202.00824/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- Builds on 33 of its 94 references that are in the map.
- First posted on arXiv 2 Feb 2022.

## Builds on (most cited first, 30 of 33)

- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [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)
- [Learning Deep Kernels for Non-Parametric Two-Sample Tests](https://mlsanity.com/paper/2002.09116/) (2020)
- [Measuring Sample Quality with Kernels](https://mlsanity.com/paper/1703.01717/) (2017)
- [Generative Models and Model Criticism via Optimized Maximum Mean Discrepancy](https://mlsanity.com/paper/1611.04488/) (2016)
- [Measuring Sample Quality with Stein's Method](https://mlsanity.com/paper/1506.03039/) (2015)
- [MMD Aggregated Two-Sample Test](https://mlsanity.com/paper/2110.15073/) (2021)
- [Minimum Stein Discrepancy Estimators](https://mlsanity.com/paper/1906.08283/) (2019)
- [Kernel Stein Discrepancy Descent](https://mlsanity.com/paper/2105.09994/) (2021)
- [A Linear-Time Kernel Goodness-of-Fit Test](https://mlsanity.com/paper/1705.07673/) (2017)
- [Blindness of score-based methods to isolated components and mixing proportions](https://mlsanity.com/paper/2008.10087/) (2020)
- [A Wild Bootstrap for Degenerate Kernel Tests](https://mlsanity.com/paper/1408.5404/) (2014)
- [A Kernel Two-Sample Test for Functional Data](https://mlsanity.com/paper/2008.11095/) (2020)
- [Random Feature Stein Discrepancies](https://mlsanity.com/paper/1806.07788/) (2018)
- [A Witness Two-Sample Test](https://mlsanity.com/paper/2102.05573/) (2021)
- [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)
- [Testing Goodness of Fit of Conditional Density Models with Kernels](https://mlsanity.com/paper/2002.10271/) (2020)
- [Learning Kernel Tests Without Data Splitting](https://mlsanity.com/paper/2006.02286/) (2020)
- [A Kernel Stein Test for Comparing Latent Variable Models](https://mlsanity.com/paper/1907.00586/) (2019)
- [Composite Goodness-of-fit Tests with Kernels](https://mlsanity.com/paper/2111.10275/) (2021)
- [A Stein Goodness-of-fit Test for Directional Distributions](https://mlsanity.com/paper/2002.06843/) (2020)
- [Kernelized Stein Discrepancy Tests of Goodness-of-fit for Time-to-Event Data](https://mlsanity.com/paper/2008.08397/) (2020)
- [Post Selection Inference with Incomplete Maximum Mean Discrepancy Estimator](https://mlsanity.com/paper/1802.06226/) (2018)
- [A maximum-mean-discrepancy goodness-of-fit test for censored data](https://mlsanity.com/paper/1810.04286/) (2018)
- [Informative Features for Model Comparison](https://mlsanity.com/paper/1810.11630/) (2018)
- [Kernel Stein Tests for Multiple Model Comparison](https://mlsanity.com/paper/1910.12252/) (2019)
- [Interpretable Stein Goodness-of-fit Tests on Riemannian Manifolds](https://mlsanity.com/paper/2103.00895/) (2021)

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

- [MMD-FUSE: Learning and Combining Kernels for Two-Sample Testing Without Data Splitting](https://mlsanity.com/paper/2306.08777/) (2023)
- [Efficient Aggregated Kernel Tests using Incomplete U-statistics](https://mlsanity.com/paper/2206.09194/) (2022)
- [Practical Kernel Tests of Conditional Independence](https://mlsanity.com/paper/2402.13196/) (2024)
- [A Practical Introduction to Kernel Discrepancies: MMD, HSIC & KSD](https://mlsanity.com/paper/2503.04820/) (2025)
- [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)
- [DUAL: Learning Diverse Kernels for Aggregated Two-sample and Independence Testing](https://mlsanity.com/paper/2510.11140/) (2025)
- [A Unified View of Optimal Kernel Hypothesis Testing](https://mlsanity.com/paper/2503.07084/) (2025)
- [Using Perturbation to Improve Goodness-of-Fit Tests based on Kernelized Stein Discrepancy](https://mlsanity.com/paper/2304.14762/) (2023)
- [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)
- [A general framework for the analysis of kernel-based tests](https://mlsanity.com/paper/2209.00124/) (2022)
- [On RKHS Choices for Assessing Graph Generators via Kernel Stein Statistics](https://mlsanity.com/paper/2210.05746/) (2022)
- [Optimal Online Change Detection via Random Fourier Features](https://mlsanity.com/paper/2505.17789/) (2025)
- [Kernel Tests of Equivalence](https://mlsanity.com/paper/2603.10886/) (2026)
- [Minimax Lower Bounds of Kernel Discrepancy Estimation: MMD, HSIC, KSD](https://mlsanity.com/paper/2607.24235/) (2026)
- [A Kernelised Stein Statistic for Assessing Implicit Generative Models](https://mlsanity.com/paper/2206.00149/) (2022)
- [Sequential Kernelized Stein Discrepancy](https://mlsanity.com/paper/2409.17505/) (2024)
- [Stein Discrepancy for Unsupervised Domain Adaptation](https://mlsanity.com/paper/2502.03587/) (2025)
- [Boosting Statistic Learning with Synthetic Data from Pretrained Large Models](https://mlsanity.com/paper/2505.04992/) (2025)
- [One-shot Conditional Sampling: MMD meets Nearest Neighbors](https://mlsanity.com/paper/2509.25507/) (2025)

## Abstract (arXiv)

We investigate properties of goodness-of-fit tests based on the Kernel Stein Discrepancy (KSD). We introduce a strategy to construct a test, called KSDAgg, which aggregates multiple tests with different kernels. KSDAgg avoids splitting the data to perform kernel selection (which leads to a loss in test power), and rather maximises the test power over a collection of kernels. We provide non-asymptotic guarantees on the power of KSDAgg: we show it achieves the smallest uniform separation rate of the collection, up to a logarithmic term. For compactly supported densities with bounded model score function, we derive the rate for KSDAgg over restricted Sobolev balls; this rate corresponds to the minimax optimal rate over unrestricted Sobolev balls, up to an iterated logarithmic term. KSDAgg can be computed exactly in practice as it relies either on a parametric bootstrap or on a wild bootstrap to estimate the quantiles and the level corrections. In particular, for the crucial choice of bandwidth of a fixed kernel, it avoids resorting to arbitrary heuristics (such as median or standard deviation) or to data splitting. We find on both synthetic and real-world data that KSDAgg outperforms other state-of-the-art quadratic-time adaptive KSD-based goodness-of-fit testing procedures.