Sobolev Regularized MMD Gradient Flow
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866909035623612416 |
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| author | Tian, Chenyang Sriperumbudur, Bharath K. Gretton, Arthur Chen, Zonghao |
| author_facet | Tian, Chenyang Sriperumbudur, Bharath K. Gretton, Arthur Chen, Zonghao |
| contents | 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 \emph{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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_11884 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sobolev Regularized MMD Gradient Flow Tian, Chenyang Sriperumbudur, Bharath K. Gretton, Arthur Chen, Zonghao Machine Learning 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 \emph{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. |
| title | Sobolev Regularized MMD Gradient Flow |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2605.11884 |