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.

翻译：在数值积分中，使用有限点集逼近目标概率分布是一个具有基础重要性的问题。多位学者曾提出通过最小化最大均值差异（MMD）来选取点集，但该目标的非凸性通常阻碍了全局最小化的实现。为此，我们考虑\emph{MMD平稳点}的概念——与全局最小化MMD的点不同，这类点能够被精确计算。我们的主要贡献具有双重理论性质。首先，我们证明了一个（或许令人惊讶的）结果：对于关联再生核希尔伯特空间中的被积函数，平稳MMD点的数值积分误差以\emph{快于}MMD本身的速度收敛至零。受此\emph{超收敛}性质的启发，我们将MMD梯度流视为计算MMD平稳点的实用策略。随后，通过建立新颖的非渐近有限粒子误差界进行精细收敛分析，我们证明了MMD梯度流确实能够计算平稳MMD点。