Stationary MMD Points

Zonghao Chen, Toni Karvonen, Heishiro Kanagawa, Francois-Xavier Briol, Chris J. Oates
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:14884-14912, 2026.

Abstract

Approximation of a target probability distribution using a finite set of points is a problem of fundamental importance in numerical integration. Several authors have proposed to select points by minimising a maximum mean discrepancy (MMD), but the non-convexity of this objective typically precludes global minimisation. Instead, we consider the concept of stationary points of the MMD which, in contrast to points globally minimising the MMD, can be accurately computed. Our main contributions are two-fold and theoretical in nature. We first prove the (perhaps surprising) result that, for integrands in the associated reproducing kernel Hilbert space, the numerical integration error of stationary MMD points vanishes faster than the MMD. Motivated by this super-convergence property, we consider MMD gradient flows as a practical strategy for computing stationary points of the MMD. We then prove that MMD gradient flow can indeed compute stationary MMD points, based on a refined convergence analysis that establishes a novel non-asymptotic finite-particle error bound.

Cite this Paper

BibTeX
@InProceedings{pmlr-v306-chen26be, title = {Stationary {MMD} Points}, author = {Chen, Zonghao and Karvonen, Toni and Kanagawa, Heishiro and Briol, Francois-Xavier and Oates, Chris J.}, booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, pages = {14884--14912}, year = {2026}, editor = {Zhang, Tong and Dudik, Miroslav and Jaggi, Martin and Agarwal, Alekh and Li, Sharon and Schuurmans, Dale and Zhu, Jerry and Berkenkamp, Felix and Dong, Hanze and Bietti, Alberto}, volume = {306}, series = {Proceedings of Machine Learning Research}, month = {06--11 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v306/main/assets/chen26be/chen26be.pdf}, url = {https://proceedings.mlr.press/v306/chen26be.html}, abstract = {Approximation of a target probability distribution using a finite set of points is a problem of fundamental importance in numerical integration. Several authors have proposed to select points by minimising a maximum mean discrepancy (MMD), but the non-convexity of this objective typically precludes global minimisation. Instead, we consider the concept of stationary points of the MMD which, in contrast to points globally minimising the MMD, can be accurately computed. Our main contributions are two-fold and theoretical in nature. We first prove the (perhaps surprising) result that, for integrands in the associated reproducing kernel Hilbert space, the numerical integration error of stationary MMD points vanishes faster than the MMD. Motivated by this super-convergence property, we consider MMD gradient flows as a practical strategy for computing stationary points of the MMD. We then prove that MMD gradient flow can indeed compute stationary MMD points, based on a refined convergence analysis that establishes a novel non-asymptotic finite-particle error bound.} }
Endnote
%0 Conference Paper %T Stationary MMD Points %A Zonghao Chen %A Toni Karvonen %A Heishiro Kanagawa %A Francois-Xavier Briol %A Chris J. Oates %B Proceedings of the 43rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2026 %E Tong Zhang %E Miroslav Dudik %E Martin Jaggi %E Alekh Agarwal %E Sharon Li %E Dale Schuurmans %E Jerry Zhu %E Felix Berkenkamp %E Hanze Dong %E Alberto Bietti %F pmlr-v306-chen26be %I PMLR %P 14884--14912 %U https://proceedings.mlr.press/v306/chen26be.html %V 306 %X Approximation of a target probability distribution using a finite set of points is a problem of fundamental importance in numerical integration. Several authors have proposed to select points by minimising a maximum mean discrepancy (MMD), but the non-convexity of this objective typically precludes global minimisation. Instead, we consider the concept of stationary points of the MMD which, in contrast to points globally minimising the MMD, can be accurately computed. Our main contributions are two-fold and theoretical in nature. We first prove the (perhaps surprising) result that, for integrands in the associated reproducing kernel Hilbert space, the numerical integration error of stationary MMD points vanishes faster than the MMD. Motivated by this super-convergence property, we consider MMD gradient flows as a practical strategy for computing stationary points of the MMD. We then prove that MMD gradient flow can indeed compute stationary MMD points, based on a refined convergence analysis that establishes a novel non-asymptotic finite-particle error bound.
APA
Chen, Z., Karvonen, T., Kanagawa, H., Briol, F. & Oates, C.J.. (2026). Stationary MMD Points. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:14884-14912 Available from https://proceedings.mlr.press/v306/chen26be.html.

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