# Sobolev Regularized MMD Gradient Flow

arXiv: https://arxiv.org/abs/2605.11884 · HTML: https://mlsanity.com/paper/2605.11884/

## 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 (most cited first, 30 of 46)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm](https://mlsanity.com/paper/1608.04471/) (2016)
- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [A Kernelized Stein Discrepancy for Goodness-of-fit Tests and Model Evaluation](https://mlsanity.com/paper/1602.03253/) (2016)
- [How to train your neural ODE: the world of Jacobian and kinetic regularization](https://mlsanity.com/paper/2002.02798/) (2020)
- [Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices](https://mlsanity.com/paper/1903.08568/) (2019)
- [A Kernel Test of Goodness of Fit](https://mlsanity.com/paper/1602.02964/) (2016)
- [Maximum Mean Discrepancy Gradient Flow](https://mlsanity.com/paper/1906.04370/) (2019)
- [Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problem](https://mlsanity.com/paper/1802.08089/) (2018)
- [Measuring Sample Quality with Kernels](https://mlsanity.com/paper/1703.01717/) (2017)
- [Stabilizing Training of Generative Adversarial Networks through Regularization](https://mlsanity.com/paper/1705.09367/) (2017)
- [Sobolev Norm Learning Rates for Regularized Least-Squares Algorithm](https://mlsanity.com/paper/1702.07254/) (2017)
- [On the geometry of Stein variational gradient descent](https://mlsanity.com/paper/1912.00894/) (2019)
- [SVGD as a kernelized Wasserstein gradient flow of the chi-squared divergence](https://mlsanity.com/paper/2006.02509/) (2020)
- [A Non-Asymptotic Analysis for Stein Variational Gradient Descent](https://mlsanity.com/paper/2006.09797/) (2020)
- [Kernel Stein Discrepancy Descent](https://mlsanity.com/paper/2105.09994/) (2021)
- [Reverse Diffusion Monte Carlo](https://mlsanity.com/paper/2307.02037/) (2023)
- [Generative Sliced MMD Flows with Riesz Kernels](https://mlsanity.com/paper/2305.11463/) (2023)
- [Towards a Complete Analysis of Langevin Monte Carlo: Beyond Poincar'e Inequality](https://mlsanity.com/paper/2303.03589/) (2023)
- [On gradient regularizers for MMD GANs](https://mlsanity.com/paper/1805.11565/) (2018)
- [Posterior Sampling Based on Gradient Flows of the MMD with Negative Distance Kernel](https://mlsanity.com/paper/2310.03054/) (2023)
- [Towards Optimal Sobolev Norm Rates for the Vector-Valued Regularized Least-Squares Algorithm](https://mlsanity.com/paper/2312.07186/) (2023)
- [On the Convergence of Gradient Descent in GANs: MMD GAN As a Gradient Flow](https://mlsanity.com/paper/2011.02402/) (2020)
- [Spectral Regularized Kernel Two-Sample Tests](https://mlsanity.com/paper/2212.09201/) (2022)
- [Regularized Stein Variational Gradient Flow](https://mlsanity.com/paper/2211.07861/) (2022)
- [Deep MMD Gradient Flow without adversarial training](https://mlsanity.com/paper/2405.06780/) (2024)
- [Wasserstein Steepest Descent Flows of Discrepancies with Riesz Kernels](https://mlsanity.com/paper/2211.01804/) (2022)

## Built on by (most cited first, 6 of 6)

- [Three-Body Scattering for Generative Modeling](https://mlsanity.com/paper/2607.18198/) (2026)
- [Thinned Mean Field Langevin Dynamics](https://mlsanity.com/paper/2605.28589/) (2026)
- [A Gradient Flow Perspective on Minimum MMD Estimation](https://mlsanity.com/paper/2607.03871/) (2026)
- [Amortised Inference through One-Step Implicit Sampling](https://mlsanity.com/paper/2610.12679/) (2026)
- [Markov Chain Convergence in Kernel-Induced Geometry](https://mlsanity.com/paper/2610.13916/) (2026)
- [On the Optimisation Landscape of Overparameterised Weighted MMD](https://mlsanity.com/paper/2610.19861/) (2026)

## 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.