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Maximum Mean Discrepancy Gradient Flow

arXiv 1906.04370 ↗First posted 11 Jun 2019stat.ML cs.LGcited by 123

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Abstract (arXiv)

We construct a Wasserstein gradient flow of the maximum mean discrepancy (MMD) and study its convergence properties. The MMD is an integral probability metric defined for a reproducing kernel Hilbert space (RKHS), and serves as a metric on probability measures for a sufficiently rich RKHS. We obtain conditions for convergence of the gradient flow towards a global optimum, that can be related to particle transport when optimizing neural networks. We also propose a way to regularize this MMD flow, based on an injection of noise in the gradient. This algorithmic fix comes with theoretical and empirical evidence. The practical implementation of the flow is straightforward, since both the MMD and its gradient have simple closed-form expressions, which can be easily estimated with samples.

Key facts. Cited by 123 papers in the map (data through 7 Oct 2026). First posted on arXiv 11 Jun 2019.

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 14; 2022: 9; 2023: 26; 2024: 25; 2025: 27; 2026: 22.

123 papers.

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  1. 2021Deep Generative Learning via Schr"odinger Bridgecited by 103
  2. 2021Large-Scale Wasserstein Gradient Flowscited by 76
  3. 2021Kernel Stein Discrepancy Descentcited by 57
  4. 2021Optimizing Functionals on the Space of Probabilities with Input Convex Neural Networkscited by 53
  5. 2022Understanding and Mitigating Overfitting in Prompt Tuning for Vision-Language Modelscited by 40
  6. 2021KALE Flow: A Relaxed KL Gradient Flow for Probabilities with Disjoint Supportcited by 37
  7. 2023Generative Sliced MMD Flows with Riesz Kernelscited by 36
  8. 2022Mirror Descent with Relative Smoothness in Measure Spaces, with application to Sinkhorn and EMcited by 33
  9. 2023A Rigorous Link between Deep Ensembles and (Variational) Bayesian Methodscited by 31
  10. 2022Birth-death dynamics for sampling: Global convergence, approximations and their asymptoticscited by 30
  11. 2023A mean-field games laboratory for generative modelingcited by 28
  12. 2023Posterior Sampling Based on Gradient Flows of the MMD with Negative Distance Kernelcited by 25
38 more
  1. 2021A Neural Tangent Kernel Perspective of GANscited by 21
  2. 2021Efficient Gradient Flows in Sliced-Wasserstein Spacecited by 21
  3. 2023Unifying GANs and Score-Based Diffusion as Generative Particle Modelscited by 21
  4. 2023Convergence of mean-field Langevin dynamics: Time and space discretization, stochastic gradient, and variance reductioncited by 19
  5. 2024Deep MMD Gradient Flow without adversarial trainingcited by 18
  6. 2023Gaussian Interpolation Flowscited by 17
  7. 2021Stein Variational Gradient Descent: many-particle and long-time asymptoticscited by 15
  8. 2021Neural Tangent Kernel Maximum Mean Discrepancycited by 15
  9. 2026Gradient Flow Drifting: Generative Modeling via Wasserstein Gradient Flows of KDE-Approximated Divergencescited by 15
  10. 2024Learning Associative Memories with Gradient Descentcited by 14
  11. 2024A Practical Guide to Sample-based Statistical Distances for Evaluating Generative Models in Sciencecited by 14
  12. 2023Towards Understanding the Dynamics of Gaussian-Stein Variational Gradient Descentcited by 13
  13. 2024Nonparametric Teaching of Implicit Neural Representationscited by 13
  14. 2024Theoretical Guarantees for Variational Inference with Fixed-Variance Mixture of Gaussianscited by 11
  15. 2024Stein transport for Bayesian inferencecited by 11
  16. 2025Scale-wise Distillation of Diffusion Modelscited by 11
  17. 2025Integral Imprecise Probability Metricscited by 11
  18. 2026Drifting Fields are not Conservativecited by 11
  19. 2023Dynamic Flows on Curved Space Generated by Labeled Datacited by 10
  20. 2025Weighted quantization using MMD: From mean field to mean shift via gradient flowscited by 10
  21. 2023Neural Wasserstein Gradient Flows for Maximum Mean Discrepancies with Riesz Kernelscited by 9
  22. 2023The Score-Difference Flow for Implicit Generative Modelingcited by 9
  23. 2023Nonparametric Iterative Machine Teachingcited by 9
  24. 2023GAD-PVI: A General Accelerated Dynamic-Weight Particle-Based Variational Inference Frameworkcited by 9
  25. 2025A Dictionary of Closed-Form Kernel Mean Embeddingscited by 9
  26. 2021Generalization Error of GAN from the Discriminator's Perspectivecited by 8
  27. 2021Dual Training of Energy-Based Models with Overparametrized Shallow Neural Networkscited by 8
  28. 2022Lipschitz-regularized gradient flows and generative particle algorithms for high-dimensional scarce datacited by 8
  29. 2023Coin Sampling: Gradient-Based Bayesian Inference without Learning Ratescited by 8
  30. 2024Neural Sinkhorn Gradient Flowcited by 8
  31. 2023Bayesian Posterior Perturbation Analysis with Integral Probability Metricscited by 7
  32. 2023Nonparametric Teaching for Multiple Learnerscited by 7
  33. 2024Fast Kernel Summation in High Dimensions via Slicing and Fourier Transformscited by 7
  34. 2024Interaction-Force Transport Gradient Flowscited by 7
  35. 2024Particle Semi-Implicit Variational Inferencecited by 7
  36. 2024Importance Corrected Neural JKO Samplingcited by 7
  37. 2024Wasserstein Gradient Flows of MMD Functionals with Distance Kernel and Cauchy Problems on Quantile Functionscited by 7
  38. 2024Fast Summation of Radial Kernels via QMC Slicingcited by 7

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