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

arXiv:2605.17963 · math.OC, stat.ML · Submitted 2026-05-18 · Read on arXiv

math.OC, stat.ML

Submitted: 2026-05-18

Updated: 2026-09-26

Comments: Major revision: corrected and refined main theoretical results, leading to updates in the title, abstract, and core contributions. Restructured the main body and added new experiments

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Terminology

• Gradient Flow — 19× in this paper · explained in 2 episodes
• Optimal Transport — 18× in this paper · explained in 5 episodes
• Wasserstein Saddle-Free Newton — 11× in this paper
• Regularization — 9× in this paper · explained in 3 episodes
• Wasserstein Distance — 8× in this paper · explained in 4 episodes
• Trace — 6× in this paper · explained in 2 episodes
• Convergence Guarantees — 4× in this paper · explained in 3 episodes
• Generative Modeling — 4× in this paper · explained in 3 episodes
• Langevin Dynamics — 4× in this paper · explained in 2 episodes
• Maximum Mean Discrepancy — 4× in this paper
• Activation Function — 3× in this paper · explained in 1 episode
• Theoretical Guarantees — 3× in this paper · explained in 2 episodes
• Hilbert Space — 2× in this paper · explained in 2 episodes
• Objective Function — 2× in this paper · explained in 2 episodes
• Reinforcement Learning — 2× in this paper · explained in 4 episodes
• Transformer — 2× in this paper · explained in 2 episodes
• Wasserstein Gradient Flow (WGF) — 2× in this paper · explained in 2 episodes
• Accuracy — 1× in this paper · explained in 1 episode
• Actor — 1× in this paper · explained in 3 episodes
• Consistency — 1× in this paper · explained in 3 episodes
• Foundation Model — 1× in this paper · explained in 3 episodes
• Foundation Models — 1× in this paper · explained in 7 episodes
• Gaussian Processes (GPs) — 1× in this paper · explained in 2 episodes
• Generalization — 1× in this paper · explained in 13 episodes
• Generative Adversarial Network — 1× in this paper
• Neural Network Approximation — 1× in this paper · explained in 2 episodes
• Uncertainty Quantification — 1× in this paper · explained in 8 episodes
• alpha — 1× in this paper · explained in 1 episode

Sources

• Quantitative Convergence of Wasserstein Gradient Flows of Kernel Mean Discrepancies
• Convergence of drift-diffusion PDEs arising as Wasserstein gradient flows of convex functions
• Generative Modeling by Minimizing the Wasserstein-2 Loss
• Linear convergence of proximal descent schemes on the Wasserstein space
• Convex Regularization and Convergence of Policy Gradient Flows under Safety Constraints
• Convex Analysis of the Mean Field Langevin Dynamics
• Information Newton's flow: second-order optimization method in probability space
• Mean-field Variational Inference via Wasserstein Gradient Flow
• Convergence Analysis of the Wasserstein Proximal Algorithm beyond Geodesic Convexity

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