arXiv Machine Learning
Jun 30

Learning from samples: inverse problems over measures

arXiv:2505. 07124v3 Announce Type: replace Abstract: We study inverse problems where an unknown potential is observed only through samples from the measure it induces by a convex variational principle.

By Francisco Andrade, Gabriel Peyr\'e, Clarice Poon
arXiv AI
Jun 24

The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space

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 Statistics ML
Aug 27

Continuum-marginal optimal transport: a mesh-free kernel method

The paper introduces a mesh‑free kernel method for continuum‑marginal optimal transport, aiming to recover the minimum‑energy velocity field that reproduces a continuous family of probability marginals. By embedding the weak continuity equation into a reproducing kernel Hilbert space, the authors obtain a sample‑only objective that eliminates spatial discretization. The velocity is represented via a linear‑in‑parameters dictionary or neural network and optimized with mini‑batch stochastic techniques, achieving accurate drift recovery and marginal consistency in synthetic experiments, and the framework also extends to the Nelson problem of stochastic optimal transport.

By Yumiharu Nakano
arXiv Machine Learning
Jun 5

Variational Entropic Optimal Transport

arXiv:2602. 02241v2 Announce Type: replace Abstract: Entropic optimal transport (EOT) in continuous spaces with quadratic cost is a classical tool for solving the domain translation problem.

By Roman Dyachenko, Nikita Gushchin, Kirill Sokolov, Petr Mokrov, Evgeny Burnaev, Alexander Korotin