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[2505.20754] Stationary MMD Points
Summary
The paper discusses the concept of stationary MMD points in numerical integration, demonstrating their advantages over traditional methods in minimizing maximum mean discrepancy (MMD).
Why It Matters
This research addresses a significant challenge in numerical integration by providing a method to compute stationary MMD points, which can lead to improved accuracy in approximating target distributions. The findings could influence future methodologies in machine learning and statistical analysis.
Key Takeaways
- Stationary MMD points can be computed more accurately than globally minimizing MMD points.
- The numerical integration error for stationary MMD points decreases faster than that of MMD.
- MMD gradient flows are proposed as a practical strategy for finding stationary MMD points.
Statistics > Machine Learning arXiv:2505.20754 (stat) [Submitted on 27 May 2025 (v1), last revised 14 Feb 2026 (this version, v2)] Title:Stationary MMD Points Authors:Zonghao Chen, Toni Karvonen, Heishiro Kanagawa, François-Xavier Briol, Chris. J. Oates View a PDF of the paper titled Stationary MMD Points, by Zonghao Chen and 4 other authors View PDF HTML (experimental) 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 \emph{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 \emph{faster} than the MMD. Motivated by this \emph{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. Subjects: Machine L...
