用平稳点优化积分精度，突破传统采样方法极限。
Stationary MMD Points
- 通过求解MMD的平稳点替代全局最小化，实现高效计算。
- 在再生核希尔伯特空间中，积分误差收敛速度超过MMD本身。
- 提出梯度流算法并证明其可精准计算平稳点，适合高维数值积分场景。
利用有限点集近似目标概率分布是数值积分中的基础问题。尽管已有研究通过最小化最大均值差异（MMD）选择点，但该目标函数的非凸性通常导致无法实现全局最小化。本文提出考虑MMD的 extit{平稳点}——与全局最小化点不同，这类点可被精确计算。主要贡献为理论性：首先证明，在关联再生核希尔伯特空间中的被积函数下，平稳MMD点的数值积分误差收敛速度 extit{快于}MMD本身，展现出罕见的 extit{超收敛}特性；基于此，我们提出以MMD梯度流作为计算平稳点的实际策略，并通过精细化的收敛分析，建立了一个新颖的非渐近有限粒子误差界，严格证明了该方法的有效性。
原文摘要 · Abstract (English)
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.
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