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 square root Hessian operator. The dynamics exploit curvature magnitude while maintaining negative curvature repulsion and positive curvature attraction, avoiding key Wasserstein Newton limitations. 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 δ improves from tilde O(δ-4) for prior perturbed first-order Wasserstein methods to tilde O(δ-3). For benign landscapes, this implies proximity to 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.