mlsanityData to 7 Oct 2026

Metrizing Weak Convergence with Maximum Mean Discrepancies

arXiv 2006.09268 ↗First posted 16 Jun 2020cs.LG math.PR math.STcited by 35

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Abstract (arXiv)

This paper characterizes the maximum mean discrepancies (MMD) that metrize the weak convergence of probability measures for a wide class of kernels. More precisely, we prove that, on a locally compact, non-compact, Hausdorff space, the MMD of a bounded continuous Borel measurable kernel k, whose reproducing kernel Hilbert space (RKHS) functions vanish at infinity, metrizes the weak convergence of probability measures if and only if k is continuous and integrally strictly positive definite (i.s.p.d.) over all signed, finite, regular Borel measures. We also correct a prior result of Simon-Gabriel & Sch"olkopf (JMLR, 2018, Thm.12) by showing that there exist both bounded continuous i.s.p.d. kernels that do not metrize weak convergence and bounded continuous non-i.s.p.d. kernels that do metrize it.

Key facts. Cited by 35 papers in the map (data through 7 Oct 2026). First posted on arXiv 16 Jun 2020.

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Citing papers per year: 2021: 6; 2022: 8; 2023: 4; 2024: 6; 2025: 7; 2026: 4.

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  1. 2024Deep Generative Models through the Lens of the Manifold Hypothesis: A Survey and New Connectionscited by 47
  2. 2021Reproducing Kernel Hilbert Space, Mercer's Theorem, Eigenfunctions, Nystr"om Method, and Use of Kernels in Machine Learning: Tutorial and Surveycited by 36
  3. 2022Evaluating generative models in high energy physicscited by 31
  4. 2021Generalized Kernel Thinningcited by 29
  5. 2024DySLIM: Dynamics Stable Learning by Invariant Measure for Chaotic Systemscited by 29
  6. 2022Regularized Stein Variational Gradient Flowcited by 18
  7. 2022Targeted Separation and Convergence with Kernel Discrepanciescited by 15
  8. 2021Controlling Wasserstein Distances by Kernel Norms with Application to Compressive Statistical Learningcited by 13
  9. 2024Lecture notes on rough paths and applications to machine learningcited by 13
  10. 2025Integral Imprecise Probability Metricscited by 11
  11. 2022Learning Counterfactually Invariant Predictorscited by 10
  12. 2023The Signature Kernelcited by 8
23 more
  1. 2022Characteristic kernels on Hilbert spaces, Banach spaces, and on sets of measurescited by 7
  2. 2021Sequential Kernel Embedding for Mediated and Time-Varying Dose Response Curvescited by 5
  3. 2022Geometric Methods for Sampling, Optimisation, Inference and Adaptive Agentscited by 5
  4. 2023Confidence and Uncertainty Assessment for Distributional Random Forestscited by 5
  5. 2022Kernel methods for long term dose response curvescited by 4
  6. 2024On the Consistency of Kernel Methods with Dependent Observationscited by 3
  7. 2025Doubly-Robust Estimation of Counterfactual Policy Mean Embeddingscited by 3
  8. 2026Minimum Distance Summaries for Robust Neural Posterior Estimationcited by 3
  9. 2021A Note on Optimizing Distributions using Kernel Mean Embeddingscited by 2
  10. 2023MMD-based Variable Importance for Distributional Random Forestcited by 2
  11. 2026Kernel Tests of Equivalencecited by 2
  12. 2021Generalized Kernel Ridge Regression for Causal Inference with Missing-at-Random Sample Selectioncited by 1
  13. 2025Towards Scalable Topological Regularizerscited by 1
  14. 2025kNNSampler: Stochastic Imputations for Recovering Missing Value Distributionscited by 1
  15. 2022Robust Flow-based Conformal Inference (FCI) with Statistical Guaranteecited by 0
  16. 2023Arbitrary Distributions Mapping via SyMOT-Flow: A Flow-based Approach Integrating Maximum Mean Discrepancy and Optimal Transportcited by 0
  17. 2024Global optimality under amenable symmetry constraintscited by 0
  18. 2024Multi-Source Unsupervised Domain Adaptation with Prototype Aggregationcited by 0
  19. 2025Kernel Mean Embedding Topology: Weak and Strong Forms for Stochastic Kernels and Implications for Model Learningcited by 0
  20. 2025Sampling as Bandits: Evaluation-Efficient Design for Black-Box Densitiescited by 0
  21. 2025Copula-Stein Discrepancy: A Generator-Based Stein Operator for Archimedean Dependencecited by 0
  22. 2026A Generative Sampler for distributions with possible discrete parameter based on Reversibilitycited by 0
  23. 2026A Gradient Flow Perspective on Minimum MMD Estimationcited by 0