Approximation of a target probability distribution using a finite set of points is a problem of fundamental importance, arising in cubature, data compression, and optimisation. Several authors have proposed to select points by minimising a maximum mean discrepancy (MMD), but the non-convexity of this objective precludes global minimisation in general. Instead, we consider \emph{stationary} points of the MMD which, in contrast to points globally minimising the MMD, can be accurately computed. Our main theoretical contribution is the (perhaps surprising) result that, for integrands in the associated reproducing kernel Hilbert space, the cubature error of stationary MMD points vanishes \emph{faster} than the MMD. Motivated by this \emph{super-convergence} property, we consider discretised gradient flows as a practical strategy for computing stationary points of the MMD, presenting a refined convergence analysis that establishes a novel non-asymptotic finite-particle error bound, which may be of independent interest.

翻译：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，出现在求积、数据压缩和优化中。多位学者曾提出通过最小化最大均值差异（MMD）来选择点集，但该目标函数的非凸性通常阻碍了全局最小化的实现。为此，我们考虑MMD的平稳点——与全局最小化MMD的点不同，这类点可以精确计算。我们的主要理论贡献是（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点的求积误差以比MMD更快的速度消失。受这一超收敛性质的启发，我们将离散化梯度流视为计算MMD平稳点的实用策略，并通过精细化收敛分析建立了一个新颖的非渐近有限粒子误差界，该结果可能具有独立的理论价值。