The paper presents a convergence analysis of the Wasserstein proximal algorithm without requiring geodesic convexity of the objective functional. By assuming a Wasserstein analog of the Euclidean Polyak–Łojasiewicz inequality, the authors prove that the algorithm achieves unbiased, linear convergence, improving on previous rates that required strong geodesic convexity. The analysis is extended to inexact proximal steps for geodesically semiconvex objectives, and numerical experiments show faster convergence than noisy gradient descent on mean‑field neural networks.
By Shuailong Zhu, Xiaohui Chen
arXiv:2605. 26078v3 Announce Type: replace Abstract: Wasserstein policy gradient (WPG) is a policy optimization method for reinforcement learning (RL) that exploits the optimal-transport geometry of action distributions.
By Zhaoyu Zhu, Rui Gao, Shuang Li
The review explores how control theory, optimal transport, probabilistic inference, non‑equilibrium thermodynamics, and machine learning are interconnected through the optimization of free‑energy‑like functionals under dynamical or statistical constraints. It presents a conceptual thread linking these five fields and illustrates the ideas with applications in reinforcement learning, variational inference, and generative modeling. The article is written for readers without prior familiarity, beginning with physics principles.
By Emmy Blumenthal, Nikolas Claussen, Benjamin Eysenbach, Catherine Ji, Gautam Reddy, Colin Scheibner, Benjamin Sorkin
arXiv:2102. 09235v3 Announce Type: replace Abstract: Recent studies revealed the mathematical connection between deep neural networks (DNNs) and dynamic systems.
By Kuo Gai, Shihua Zhang
arXiv:2610.01413v1 Announce Type: cross
Abstract: Reinforcement learning (RL) algorithms frequently compare probability distributions, such as state visitation distributions induced by policies and e...
By Yujie Zhu, Charles A. Hepburn, Matthew Thorpe, Giovanni Montana
arXiv:2608. 07433v1 Announce Type: cross Abstract: Wasserstein policy gradient (WPG) updates state-conditional action laws by transport in the action space.
By Zhaoyu Zhu, Rui Gao, Shuang Li
The paper develops a diffusion approximation for stochastic gradient descent (SGD) when the optimization target is a functional on the Wasserstein space ℝ2. By lifting the problem to a Hilbert space via Lions differentiability, the authors construct a Gaussian random-field approximation whose velocity field matches the mean and covariance of the original stochastic gradient. They prove that this Gaussian approximation achieves second‑order weak accuracy, providing a rigorous basis for replacing sample‑driven randomness with analytically tractable Gaussian fluctuations in stochastic optimization over probability measures.
By Maria Oprea, Qin Li, Yunan Yang
Light Entropic Optimal Transport on Riemannian Manifolds introduces ManifoldLightOT, a lightweight method for learning kernel‑induced entropic optimal transport couplings directly on common manifolds such as spheres, tori, SO(3), and SE(3). By constructing geometry‑specific Gibbs kernels and compatible potential parameterizations, the approach yields closed‑form normalization and directly sampleable conditional distributions. The formulation extends naturally to products of manifolds and optimizes potentials from samples using Monte Carlo estimates, demonstrating superior performance over existing manifold OT methods in synthetic and real‑world experiments.
By Xavier Aramayo-Carrasco, Petr Mokrov, Alexander Korotin
arXiv:2506. 04480v2 Announce Type: replace-cross Abstract: This paper focuses on Geodesic Principal Component Analysis (GPCA) on a collection of probability distributions using the Otto-Wasserstein geometry.
By Nina Vesseron, Elsa Cazelles, Alice Le Brigant, Thierry Klein
arXiv:2606. 24157v1 Announce Type: new Abstract: The space $\mathcal{P}_2(\mathbb{R}^d$) of probability measures with finite second moment carries a natural geometry: the quadratic Wasserstein distance W_2 makes it a complete metric space and, following Otto, a (formal) Riemannian manifold whose geodesics are the optimal-transport interpolations.
By Yian Yao, Weiwei Zhang
arXiv:2606. 27767v1 Announce Type: new Abstract: Optimizing functionals over the space of probability measures is now ubiquitous in machine learning.
By Cl\'ement Bonet, Pierre-Cyril Aubin-Frankowski, Youssef Mroueh
arXiv:2505. 06589v2 Announce Type: replace-cross Abstract: Modern machine learning repeatedly manipulates probability measures: empirical datasets, generated samples, latent distributions, class-conditional laws, particle systems, weights of wide networks and attention patterns.
By Gabriel Peyr\'e