Wasserstein Saddle-Free Newton: Saddle Escape and Local Spectral Convergence

Extending saddle-free Newton methods to Wasserstein space requires controlling Hessian dependent operators along nonlinear transport trajectories and accounting for a Hessian spectrum that may accumulate at zero. We address these challenges through Wasserstein Saddle-Free Newton (WSFN), a second-order method for nonconvex optimization over Wasserstein space that preconditions the Wasserstein gradient by a regularized inverse squared Hessian operator. The resulting dynamics exploit curvature magnitude without reversing the repulsive effect of negative curvature, while retaining attraction along positive curvature directions, thereby avoiding a fundamental limitation of standard Wasserstein Newton dynamics. Whereas previous Wasserstein saddle escape results rely on perturbed first-order dynamics, WSFN uses curvature both to modify the deterministic transport and to construct the perturbation mechanism. Our analysis combines stability estimates for Hessian operators with a perturbative saddle escape argument to establish polynomial time convergence to approximate second-order stationarity under regularity of landscape assumptions. The contribution of the saddle curvature parameter $\delta$ improves from $\tilde{O}({\delta }^{-4})$ for prior perturbed first-order Wasserstein methods to $\tilde{O}({\delta }^{-3})$. For benign landscapes, this further implies proximity to the set of global minimizers. The cubic dependence corresponds to the scale of the objective decrease that can be guaranteed from negative curvature under Lipschitz Hessian regularity. Near a minimizer, the possible absence of a uniform spectral gap prevents a uniform contraction result and instead calls for a spectral characterization. We prove geometric contraction of the linearized dynamics on positive curvature spectral bands bounded away from zero and polynomial decay under spectral source conditions. Together, these results connect quantitative saddle escape with the local spectral behavior of second-order Wasserstein dynamics.