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.
arXiv:2609.25659v1 Announce Type: new Abstract: Many scientific datasets, such as molecular conformational ensembles or single-cell tissue measurements, are naturally modeled as meta-distributions: d...
arXiv:2510. 22778v3 Announce Type: replace-cross Abstract: We develop a free-probabilistic framework for denoising diffusion, in which the data is a self-adjoint operator and its law a spectral distribution.
arXiv:2606. 16610v1 Announce Type: cross Abstract: Diffusion Flow Matching (DFM) has recently emerged as a versatile framework for generative modeling, yet its theoretical convergence properties remain only partially understood.
arXiv:2410. 01244v2 Announce Type: replace-cross Abstract: We introduce a novel Wasserstein-1 ($W_1$) path-space divergence for stochastic and deterministic dynamics and establish a Wasserstein Uncertainty Propagation (WUP) theorem that bounds the $W_1$ distance between terminal distributions by the proposed divergence, equivalently characterized by a weighted $L^2$ discrepancy between the underlying drifts and the $W_1$ distance between their initial measures.
RW-Flow presents a new one‑step generative framework for data on compact manifolds, leveraging Wasserstein gradient flows. The authors derive a necessary and sufficient identifiability condition for velocity fields on compact, connected Riemannian manifolds, showing that a symmetric, Lipschitz‑continuous cost function yields identifiability iff its Gibbs kernel is nondegenerate. Experiments on geospatial events, protein and RNA torsion angles, and discretized manifolds demonstrate that RW‑Flow surpasses existing one‑step methods across most benchmark settings.
arXiv:2602. 22265v2 Announce Type: replace Abstract: Modern vision generators transport a base distribution to data through time-indexed measures, implemented as deterministic flows (ODEs) or stochastic diffusions (SDEs).
The paper introduces a geometric framework for reinforcement learning that treats policies as mappings into the Wasserstein space of action probabilities. It establishes a Riemannian structure induced by stationary distributions, defines the tangent space of policies, and characterizes geodesics while addressing measurability concerns. The authors formulate a general RL optimization problem, construct a gradient flow via Otto's calculus, compute the gradient and Hessian of the energy, and demonstrate the approach with numerical examples for low‑dimensional problems and neural‑network‑parameterized policies for high‑dimensional settings.
arXiv:2606. 17196v1 Announce Type: cross Abstract: This paper is concerned with learning principal variations of random probability measures on $\mathbb{R}^m$ under the Wasserstein geometry.
arXiv:2606. 06179v1 Announce Type: cross Abstract: Score-based diffusion models are typically trained by minimizing the $L^2$ score matching error, and standard theoretical analyses rely on this quantity to bound the sampling discrepancy between the learned and target distributions.
arXiv:2607. 06497v1 Announce Type: new Abstract: We introduce EntroPath, a manifold learning method that recovers geodesic geometry from data graphs through ensembles of diffusion paths.
arXiv:2102. 09235v3 Announce Type: replace Abstract: Recent studies revealed the mathematical connection between deep neural networks (DNNs) and dynamic systems.
arXiv:2609. 27008v1 Announce Type: cross Abstract: We study the long-time behavior of Wasserstein gradient flows for interaction energies \[ \mathsf E[\mu] = \frac12\iint_{M\times M}K(x,y)\,\mathrm d\mu(x)\,\mathrm d\mu(y) \] on a closed manifold $M$.