mlsanityData to 14 Oct 2026

A Dictionary of Closed-Form Kernel Mean Embeddings

arXiv 2504.18830 ↗First posted 26 Apr 2025stat.ML cs.LG cs.NAcited by 9

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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.

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

52 of 86 references.

References · 52

  1. 2017Wasserstein Auto-Encoderscited by 373
  2. 2016Kernel Mean Embedding of Distributions: A Review and Beyondcited by 364
  3. 2015Training generative neural networks via Maximum Mean Discrepancy optimizationcited by 151
  4. 2016A Kernel Test of Goodness of Fitcited by 143
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38 more
  1. 2019Statistical Inference for Generative Models with Maximum Mean Discrepancycited by 45
  2. 2018Stein Pointscited by 43
  3. 2022Robust Bayesian Inference for Simulator-based Models via the MMD Posterior Bootstrapcited by 41
  4. 2023Generative Sliced MMD Flows with Riesz Kernelscited by 37
  5. 2020Kernel Methods for Causal Functions: Dose, Heterogeneous, and Incremental Response Curvescited by 36
  6. 2019MMD-Bayes: Robust Bayesian Estimation via Maximum Mean Discrepancycited by 35
  7. 2018Counterfactual Mean Embeddingscited by 32
  8. 2021Stein's Method Meets Computational Statistics: A Review of Some Recent Developmentscited by 32
  9. 2014Sampling for Inference in Probabilistic Models with Fast Bayesian Quadraturecited by 31
  10. 2019Stein Point Markov Chain Monte Carlocited by 29
  11. 2019Kernel quadrature with DPPscited by 27
  12. 2015Probabilistic Integration: A Role in Statistical Computation?cited by 26
  13. 2015Sequential Kernel Herding: Frank-Wolfe Optimization for Particle Filteringcited by 22
  14. 2019Active Multi-Information Source Bayesian Quadraturecited by 17
  15. 2014Kernel Mean Shrinkage Estimatorscited by 16
  16. 2017Convergence Analysis of Deterministic Kernel-Based Quadrature Rules in Misspecified Settingscited by 16
  17. 2018Bayesian Quadrature for Multiple Related Integralscited by 16
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  19. 2016Bayesian Learning of Kernel Embeddingscited by 15
  20. 2021Vector-Valued Control Variatescited by 13
  21. 2023Optimally-Weighted Estimators of the Maximum Mean Discrepancy for Likelihood-Free Inferencecited by 13
  22. 2021Discrepancy-based Inference for Intractable Generative Models using Quasi-Monte Carlocited by 12
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  24. 2019Closed-form Expressions for Maximum Mean Discrepancy with Applications to Wasserstein Auto-Encoderscited by 11
  25. 2021Generalized Bayesian Likelihood-Free Inferencecited by 11
  26. 2023Kernel quadrature with randomly pivoted Choleskycited by 11
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  32. 2024Conditional Bayesian Quadraturecited by 9
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  35. 2022Variance-Aware Estimation of Kernel Mean Embeddingcited by 6
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  38. 2017Gaussian Process Quadrature Moment Transformcited by 3

50 most cited of 52 All 52 (JSON)

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2025: 4; 2026: 5.

9 papers.

  1. 2025Stationary MMD Pointscited by 2
  2. 2026Bayesian Quadraturecited by 1
  3. 2025Kernel Quantile Embeddings and Associated Probability Metricscited by 0
  4. 2025Bilateral Distribution Compression: Reducing Both Data Size and Dimensionalitycited by 0
  5. 2025BayesSum: Bayesian Quadrature in Discrete Spacescited by 0
  6. 2026A Distribution-to-Distribution Neural Probabilistic Forecasting Framework for Dynamical Systemscited by 0
  7. 2026Gaussian mixture models in Hilbert spaces via kernel methodscited by 0
  8. 2026Scalable Bayesian Inference for Nonlinear Conservation Lawscited by 0
  9. 2026Hierarchical Bayesian Quadraturecited by 0