# A Dictionary of Closed-Form Kernel Mean Embeddings

arXiv: https://arxiv.org/abs/2504.18830 · HTML: https://mlsanity.com/paper/2504.18830/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 52 of its 86 references that are in the map.
- First posted on arXiv 26 Apr 2025.

## Builds on (most cited first, 30 of 52)

- [Wasserstein Auto-Encoders](https://mlsanity.com/paper/1711.01558/) (2017)
- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Training generative neural networks via Maximum Mean Discrepancy optimization](https://mlsanity.com/paper/1505.03906/) (2015)
- [A Kernel Test of Goodness of Fit](https://mlsanity.com/paper/1602.02964/) (2016)
- [Maximum Mean Discrepancy Gradient Flow](https://mlsanity.com/paper/1906.04370/) (2019)
- [Kernel Instrumental Variable Regression](https://mlsanity.com/paper/1906.00232/) (2019)
- [Control functionals for Monte Carlo integration](https://mlsanity.com/paper/1410.2392/) (2014)
- [Minimum Stein Discrepancy Estimators](https://mlsanity.com/paper/1906.08283/) (2019)
- [Kernel Stein Discrepancy Descent](https://mlsanity.com/paper/2105.09994/) (2021)
- [Robust Generalised Bayesian Inference for Intractable Likelihoods](https://mlsanity.com/paper/2104.07359/) (2021)
- [Learning Robust Statistics for Simulation-based Inference under Model Misspecification](https://mlsanity.com/paper/2305.15871/) (2023)
- [Kernel Thinning](https://mlsanity.com/paper/2105.05842/) (2021)
- [Statistical Inference for Generative Models with Maximum Mean Discrepancy](https://mlsanity.com/paper/1906.05944/) (2019)
- [Stein Points](https://mlsanity.com/paper/1803.10161/) (2018)
- [Robust Bayesian Inference for Simulator-based Models via the MMD Posterior Bootstrap](https://mlsanity.com/paper/2202.04744/) (2022)
- [Generative Sliced MMD Flows with Riesz Kernels](https://mlsanity.com/paper/2305.11463/) (2023)
- [Kernel Methods for Causal Functions: Dose, Heterogeneous, and Incremental Response Curves](https://mlsanity.com/paper/2010.04855/) (2020)
- [MMD-Bayes: Robust Bayesian Estimation via Maximum Mean Discrepancy](https://mlsanity.com/paper/1909.13339/) (2019)
- [Counterfactual Mean Embeddings](https://mlsanity.com/paper/1805.08845/) (2018)
- [Stein's Method Meets Computational Statistics: A Review of Some Recent Developments](https://mlsanity.com/paper/2105.03481/) (2021)
- [Sampling for Inference in Probabilistic Models with Fast Bayesian Quadrature](https://mlsanity.com/paper/1411.0439/) (2014)
- [Stein Point Markov Chain Monte Carlo](https://mlsanity.com/paper/1905.03673/) (2019)
- [Kernel quadrature with DPPs](https://mlsanity.com/paper/1906.07832/) (2019)
- [Probabilistic Integration: A Role in Statistical Computation?](https://mlsanity.com/paper/1512.00933/) (2015)
- [Sequential Kernel Herding: Frank-Wolfe Optimization for Particle Filtering](https://mlsanity.com/paper/1501.02056/) (2015)
- [Active Multi-Information Source Bayesian Quadrature](https://mlsanity.com/paper/1903.11331/) (2019)
- [Kernel Mean Shrinkage Estimators](https://mlsanity.com/paper/1405.5505/) (2014)
- [Convergence Analysis of Deterministic Kernel-Based Quadrature Rules in Misspecified Settings](https://mlsanity.com/paper/1709.00147/) (2017)
- [Bayesian Quadrature for Multiple Related Integrals](https://mlsanity.com/paper/1801.04153/) (2018)
- [(De)-regularized Maximum Mean Discrepancy Gradient Flow](https://mlsanity.com/paper/2409.14980/) (2024)

## Built on by (most cited first, 9 of 9)

- [Stationary MMD Points](https://mlsanity.com/paper/2505.20754/) (2025)
- [Bayesian Quadrature](https://mlsanity.com/paper/2602.16218/) (2026)
- [Kernel Quantile Embeddings and Associated Probability Metrics](https://mlsanity.com/paper/2505.20433/) (2025)
- [Bilateral Distribution Compression: Reducing Both Data Size and Dimensionality](https://mlsanity.com/paper/2509.17543/) (2025)
- [BayesSum: Bayesian Quadrature in Discrete Spaces](https://mlsanity.com/paper/2512.16105/) (2025)
- [A Distribution-to-Distribution Neural Probabilistic Forecasting Framework for Dynamical Systems](https://mlsanity.com/paper/2603.25370/) (2026)
- [Gaussian mixture models in Hilbert spaces via kernel methods](https://mlsanity.com/paper/2605.05996/) (2026)
- [Scalable Bayesian Inference for Nonlinear Conservation Laws](https://mlsanity.com/paper/2605.31127/) (2026)
- [Hierarchical Bayesian Quadrature](https://mlsanity.com/paper/2607.10793/) (2026)

## Abstract (arXiv)

Kernel mean embeddings -- integrals of a kernel with respect to a probability distribution -- are essential in Bayesian quadrature, but also widely used in other computational tools for numerical integration or for statistical inference based on the maximum mean discrepancy. These methods often require, or are enhanced by, the availability of a closed-form expression for the kernel mean embedding. However, deriving such expressions can be challenging, limiting the applicability of kernel-based techniques when practitioners do not have access to a closed-form embedding. This paper addresses this limitation by providing a comprehensive dictionary of known kernel mean embeddings, along with practical tools for deriving new embeddings from known ones. We also provide a Python library that includes minimal implementations of the embeddings.