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.
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:2606. 07926v1 Announce Type: cross Abstract: Optimal transport couplings are probabilistic objects, while many learning pipelines require deterministic maps.
arXiv:2505. 04338v3 Announce Type: replace Abstract: We propose Riemannian Denoising Diffusion Probabilistic Models (RDDPMs) for learning distributions on submanifolds of Euclidean space that are level sets of functions, including most of the manifolds relevant to applications.
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:2606. 14334v1 Announce Type: new Abstract: High-dimensional datasets often concentrate near low-dimensional structures, but estimating their geometry from samples typically relies on graphs and kernels that scale poorly with dataset size and dimension.
arXiv:2602.19600v2 Announce Type: replace Abstract: Many high-dimensional datasets concentrate near a low-dimensional structure embedded in the ambient space. Generative models for such data must con...
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:2603. 10718v3 Announce Type: replace Abstract: Flow Matching enables simulation-free training of generative models on Riemannian manifolds, yet sampling typically still relies on numerically integrating a probability-flow ODE.
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.
The paper introduces a new optimal transport framework that uses the Barron energy—an intrinsic neural network complexity measure—instead of the traditional kinetic energy. By modeling transport as deep residual networks that map a reference distribution to a target distribution, the authors define a new metric on probability measures and analyze its properties for generative modeling. They demonstrate that diffusion generative models are suboptimal in this Barron geometry and explore how adaptivity can improve alternative generative approaches.
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.
The paper introduces flow charts, a conditional invertible neural network designed to approximate normal coordinates on Riemannian manifolds, enabling fast Euclidean approximations of exponential and logarithmic maps. It presents a novel nonnegative loss function that is zero exactly when the chart is normal, and demonstrates that flow charts match or surpass specialized neural geodesic interpolators across sixteen synthetic and data‑derived geometries. When pretrained, flow charts achieve a median 60–70× speedup over numerical baselines while maintaining comparable logarithmic map estimation error and improving numerical geodesic solver accuracy.