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Kernel Quantile Embeddings and Associated Probability Metrics

arXiv 2505.20433 ↗First posted 26 May 2025stat.ML cs.LG math.STcited by 0

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Abstract (arXiv)

Embedding probability distributions into reproducing kernel Hilbert spaces (RKHS) has enabled powerful nonparametric methods such as the maximum mean discrepancy (MMD), a statistical distance with strong theoretical and computational properties. At its core, the MMD relies on kernel mean embeddings to represent distributions as mean functions in RKHS. However, it remains unclear if the mean function is the only meaningful RKHS representation. Inspired by generalised quantiles, we introduce the notion of kernel quantile embeddings (KQEs). We then use KQEs to construct a family of distances that: (i) are probability metrics under weaker kernel conditions than MMD; (ii) recover a kernelised form of the sliced Wasserstein distance; and (iii) can be efficiently estimated with near-linear cost. Through hypothesis testing, we show that these distances offer a competitive alternative to MMD and its fast approximations.

Key facts. No citing papers in the map yet (data through 14 Oct 2026). Builds on 39 of its 94 references that are in the map. First posted on arXiv 26 May 2025.

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References · 39

  1. 2013Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distancescited by 2,316
  2. 2019Do ImageNet Classifiers Generalize to ImageNet?cited by 1,515
  3. 2016Kernel Mean Embedding of Distributions: A Review and Beyondcited by 364
  4. 2019Generalized Sliced Wasserstein Distancescited by 208
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  8. 2018Generative Modeling using the Sliced Wasserstein Distancecited by 108
  9. 2020A Measure-Theoretic Approach to Kernel Conditional Mean Embeddingscited by 86
  10. 2020Statistical and Topological Properties of Sliced Probability Divergencescited by 85
  11. 2021MMD Aggregated Two-Sample Testcited by 75
  12. 2015Fast Two-Sample Testing with Analytic Representations of Probability Measurescited by 53
27 more
  1. 2014Learning Theory for Distribution Regressioncited by 48
  2. 2024Model Equality Testing: Which Model Is This API Serving?cited by 47
  3. 2019Statistical Inference for Generative Models with Maximum Mean Discrepancycited by 45
  4. 2021Galaxy Zoo DECaLS: Detailed Visual Morphology Measurements from Volunteers and Deep Learning for 314,000 Galaxiescited by 41
  5. 2022Information Theory with Kernel Methodscited by 37
  6. 2019MMD-Bayes: Robust Bayesian Estimation via Maximum Mean Discrepancycited by 35
  7. 2023MMD-FUSE: Learning and Combining Kernels for Two-Sample Testing Without Data Splittingcited by 33
  8. 2018Counterfactual Mean Embeddingscited by 32
  9. 2021RKHS-SHAP: Shapley Values for Kernel Methodscited by 30
  10. 2022Efficient Aggregated Kernel Tests using Incomplete U-statisticscited by 27
  11. 2021Deconditional Downscaling with Gaussian Processescited by 24
  12. 2022Nystr"om Kernel Mean Embeddingscited by 22
  13. 2021BayesIMP: Uncertainty Quantification for Causal Data Fusioncited by 21
  14. 2020Testing Goodness of Fit of Conditional Density Models with Kernelscited by 19
  15. 2023Explaining the Uncertain: Stochastic Shapley Values for Gaussian Process Modelscited by 19
  16. 2024Credal Two-Sample Tests of Epistemic Uncertaintycited by 16
  17. 2016Bayesian Learning of Kernel Embeddingscited by 15
  18. 2023Optimally-Weighted Estimators of the Maximum Mean Discrepancy for Likelihood-Free Inferencecited by 13
  19. 2021Discrepancy-based Inference for Intractable Generative Models using Quasi-Monte Carlocited by 12
  20. 2018MONK -- Outlier-Robust Mean Embedding Estimation by Median-of-Meanscited by 9
  21. 2025A Dictionary of Closed-Form Kernel Mean Embeddingscited by 9
  22. 2022Characteristic kernels on Hilbert spaces, Banach spaces, and on sets of measurescited by 7
  23. 2024An Overview of Causal Inference using Kernel Embeddingscited by 6
  24. 2023Kernelized Cumulants: Beyond Kernel Mean Embeddingscited by 5
  25. 2025A Unified View of Optimal Kernel Hypothesis Testingcited by 5
  26. 2024Statistical and Computational Guarantees of Kernel Max-Sliced Wasserstein Distancescited by 4
  27. 2022Kernel-Based Generalized Median Computation for Consensus Learningcited by 3

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