We study the minimization of non-convex functionals over the Wasserstein space. While recent work has showed that perturbed Wasserstein gradient methods can avoid saddle points for benign landscapes, existing approaches remain essentially first-order and do not provide fast local convergence once the iterates enter a neighborhood of a global minimizer. We propose Wasserstein Saddle-Free Newton (WSFN), a second-order method that preconditions the Wasserstein gradient by a regularized square root of the squared Wasserstein Hessian. This construction preserves attraction toward directions of positive curvature while inducing repulsion along directions of negative curvature, thereby overcoming the tendency of standard Wasserstein Newton dynamics to be attracted to saddles. We also establish second-order sufficient optimality conditions on Wasserstein space for strict local minimality. Under regularity and benign landscape assumptions, we prove that WSFN escapes saddle regions and reaches an $α$-neighborhood of a global minimizer in polynomial time, with improved dependence on saddle parameters compared with prior perturbed first-order methods. Once inside this neighborhood, we show that WSFN converges linearly in $L^2$-Wasserstein distance to a non-degenerate global minimizer. Finally, we present a particle-based implementation of the method.

翻译：我们研究Wasserstein空间上非凸泛函的最小化问题。尽管近期工作表明，在良性景观下，扰动Wasserstein梯度方法能够避免鞍点，但现有方法本质上仍为一阶方法，一旦迭代进入全局最小值邻域，便无法提供快速局部收敛。我们提出Wasserstein无鞍牛顿法（WSFN），这是一种二阶方法，通过正则化的Wasserstein Hessian矩阵之平方根对Wasserstein梯度进行预处理。该构造在沿负曲率方向诱导排斥的同时，保持对正曲率方向的吸引，从而克服了标准Wasserstein牛顿动力学倾向于被鞍点吸引的特性。我们还建立了Wasserstein空间上严格局部极小性的二阶充分最优性条件。在正则性和良性景观假设下，我们证明WSFN能在多项式时间内逃离鞍点区域并到达全局最小值的α-邻域，且相较于先前的扰动一阶方法，其对鞍点参数的依赖性有所改善。一旦进入该邻域，我们证明WSFN在L²-Wasserstein距离下线性收敛至非退化全局最小值。最后，我们给出了该方法的粒子实现。