研究证明加权 MMD 优化在过参数化下无虚假局部极小值
On the Optimisation Landscape of Overparameterised Weighted MMD
理论向读者看过来：这篇论文证明了粒子够多时加权 MMD 优化不会卡在坏的局部极小值，还给了显式阈值。
一篇 arXiv 论文研究了最大均值差异（MMD）在加权原子测度上的优化景观。针对一维高斯核，作者证明只要粒子数量足够多，任何紧支撑目标分布下所有局部极小值的 MMD 都可任意小，且 MMD 不为零的驻点在梯度流下不稳定。当目标分布本身有限支撑时，论文给出了显式粒子阈值，超过该阈值即不存在虚假局部极小值。作者称这是加权 MMD 优化中首个有限粒子过参数化保证。
On the Optimisation Landscape of Overparameterised Weighted MMD
We study the optimisation landscape of the maximum mean discrepancy (MMD) over weighted atomic measures. For the one-dimensional Gaussian kernel, we show that, for any compactly supported target distribution, sufficiently many particles ensure that every local minimum has arbitrarily small MMD. Moreover, any stationary point whose MMD remains bounded away from zero is unstable under an unweighted gradient flow over the particle locations. These results hold both when the weights are constrained to the simplex, and when they are unconstrained but optimised at each step. When the target is itself finitely supported, we further strengthen this result: beyond an explicit particle threshold, there are no spurious local minima, and every spurious stationary point is unstable. To our knowledge, these are the first finite-particle overparameterisation guarantees of their kind for weighted MMD optimisation.