mlsanityData to 14 Oct 2026

Sobolev Regularized MMD Gradient Flow

arXiv 2605.11884 ↗First posted 12 May 2026cs.LGcited by 6

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Abstract (arXiv)

We propose Sobolev-regularized Maximum Mean Discrepancy (SrMMD) gradient flow, a regularized variant of maximum mean discrepancy (MMD) gradient flow based on a gradient penalty on the witness function. The proposed regularization mitigates the non-convexity of the MMD objective and yields provable global convergence guarantees in MMD in both continuous and discrete time. A more surprising appeal is that our convergence analysis does not rely on isoperimetric assumptions on the target distribution. Instead, it is based on a regularity condition on the difference between kernel mean embeddings. A key highlight of the proposed flow is that it is applicable in both sampling (from an unnormalized target distribution) -- using Stein kernels -- and generative modeling settings, unlike previous works, where a gradient flow is suitable for only generative modeling or sampling but not both. The effectiveness of the proposed flow is empirically verified on a broad range of tasks in both generative modelling and sampling.

Key facts. Cited by 6 papers in the map (data through 14 Oct 2026). Builds on 46 of its 77 references that are in the map. First posted on arXiv 12 May 2026.

Builds on

46 of 77 references.

References · 46

  1. 2020Denoising Diffusion Probabilistic Modelscited by 21,076
  2. 2020Score-Based Generative Modeling through Stochastic Differential Equationscited by 9,122
  3. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,101
  4. 2017Improved Training of Wasserstein GANscited by 2,235
  5. 2016Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithmcited by 507
  6. 2016Kernel Mean Embedding of Distributions: A Review and Beyondcited by 364
  7. 2016A Kernelized Stein Discrepancy for Goodness-of-fit Tests and Model Evaluationcited by 236
  8. 2020How to train your neural ODE: the world of Jacobian and kinetic regularizationcited by 228
  9. 2019Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Sufficescited by 227
  10. 2016A Kernel Test of Goodness of Fitcited by 143
  11. 2019Maximum Mean Discrepancy Gradient Flowcited by 127
  12. 2018Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problemcited by 123
34 more
  1. 2017Measuring Sample Quality with Kernelscited by 119
  2. 2017Stabilizing Training of Generative Adversarial Networks through Regularizationcited by 118
  3. 2017Sobolev Norm Learning Rates for Regularized Least-Squares Algorithmcited by 114
  4. 2019On the geometry of Stein variational gradient descentcited by 82
  5. 2020SVGD as a kernelized Wasserstein gradient flow of the chi-squared divergencecited by 64
  6. 2020A Non-Asymptotic Analysis for Stein Variational Gradient Descentcited by 61
  7. 2021Kernel Stein Discrepancy Descentcited by 57
  8. 2023Reverse Diffusion Monte Carlocited by 41
  9. 2023Generative Sliced MMD Flows with Riesz Kernelscited by 37
  10. 2023Towards a Complete Analysis of Langevin Monte Carlo: Beyond Poincar'e Inequalitycited by 31
  11. 2018On gradient regularizers for MMD GANscited by 28
  12. 2023Posterior Sampling Based on Gradient Flows of the MMD with Negative Distance Kernelcited by 25
  13. 2023Towards Optimal Sobolev Norm Rates for the Vector-Valued Regularized Least-Squares Algorithmcited by 22
  14. 2020On the Convergence of Gradient Descent in GANs: MMD GAN As a Gradient Flowcited by 21
  15. 2022Spectral Regularized Kernel Two-Sample Testscited by 20
  16. 2022Regularized Stein Variational Gradient Flowcited by 19
  17. 2024Deep MMD Gradient Flow without adversarial trainingcited by 19
  18. 2022Wasserstein Steepest Descent Flows of Discrepancies with Riesz Kernelscited by 16
  19. 2024(De)-regularized Maximum Mean Discrepancy Gradient Flowcited by 16
  20. 2024Wasserstein Gradient Flows for Moreau Envelopes of f-Divergences in Reproducing Kernel Hilbert Spacescited by 15
  21. 2024Improved Finite-Particle Convergence Rates for Stein Variational Gradient Descentcited by 14
  22. 2022Controlling Moments with Kernel Stein Discrepanciescited by 13
  23. 2024Optimal Rates for Vector-Valued Spectral Regularization Learning Algorithmscited by 12
  24. 2020Unbalanced Sobolev Descentcited by 11
  25. 2026Quantitative Convergence of Wasserstein Gradient Flows of Kernel Mean Discrepanciescited by 10
  26. 2024Minimax Optimal Goodness-of-Fit Testing with Kernel Stein Discrepancycited by 9
  27. 2024Conditional Bayesian Quadraturecited by 9
  28. 2022Lipschitz-regularized gradient flows and generative particle algorithms for high-dimensional scarce datacited by 8
  29. 2024Wasserstein Gradient Flows of MMD Functionals with Distance Kernel and Cauchy Problems on Quantile Functionscited by 7
  30. 2023Spectral Regularized Kernel Goodness-of-Fit Testscited by 6
  31. 2024Annealed Stein Variational Gradient Descent for Improved Uncertainty Estimation in Full-Waveform Inversioncited by 5
  32. 2025Nested Expectations with Kernel Quadraturecited by 5
  33. 2020Differential Inclusions in Wasserstein Spaces: The Cauchy-Lipschitz Frameworkcited by 4
  34. 2024Sampling from multi-modal distributions with polynomial query complexity in fixed dimension via reverse diffusioncited by 4

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2026: 6.

6 papers.

  1. 2026Three-Body Scattering for Generative Modelingcited by 4
  2. 2026Thinned Mean Field Langevin Dynamicscited by 1
  3. 2026A Gradient Flow Perspective on Minimum MMD Estimationcited by 1
  4. 2026Amortised Inference through One-Step Implicit Samplingcited by 0
  5. 2026Markov Chain Convergence in Kernel-Induced Geometrycited by 0
  6. 2026On the Optimisation Landscape of Overparameterised Weighted MMDcited by 0