← 研踪 PaperTrailarXiv:2611.15220 · v1
FréchetFlow: Flow Matching for Non-Euclidean Data
首次提交 2026-11-11 17:14
版本更新 2026-11-11 17:14
雷达观测 2026-11-13 11:11
摘要
Scientific observations such as probability distributions, trees and covariance matrices call for generative models that respect their non-Euclidean geometry. While flow matching has been extended to manifolds by utilizing geodesics and velocities, such manifold structures are not available in a general metric space. We introduce FréchetFlow, a generative modeling framework for metric space-valued data that requires neither linear operations nor a smooth manifold structure. A key feature of the proposed approach is that we utilize well-established Euclidean flows to learn the joint distribution of distances to a set of landmarks, whereupon new objects are reconstructed from generated landmark representations. The reconstruction step utilizes weighted Fréchet means of reference objects, drawing on Fréchet regression. We establish Wasserstein error bounds that connect accuracy of the generated object distributions to the placement of landmarks, flow approximation and reconstruction error and prove consistency under finite second moments. This leads to convergence rates that depend on the metric complexity of the target objects, tail moments and fitting accuracy. Experiments show competitive accuracy with established flow matching methods in Euclidean simulations and lower Wasserstein errors than specifically tailored methods for several scientific datasets. Across simulation settings, we find that generation error decreases as sample size increases.
作者备注：53 pages, 7 figures
观测到的版本
相邻记录可能跳过中间版本；首次发现 v2 及以上也会标为版本更新。
v1 · 2026-11-11 17:14 · 最新观测版本
FréchetFlow: Flow Matching for Non-Euclidean Data
Scientific observations such as probability distributions, trees and covariance matrices call for generative models that respect their non-Euclidean geometry. While flow matching has been extended to manifolds by utilizing geodesics and velocities, such manifold structures are not available in a general metric space. We introduce FréchetFlow, a generative modeling framework for metric space-valued data that requires neither linear operations nor a smooth manifold structure. A key feature of the proposed approach is that we utilize well-established Euclidean flows to learn the joint distribution of distances to a set of landmarks, whereupon new objects are reconstructed from generated landmark representations. The reconstruction step utilizes weighted Fréchet means of reference objects, drawing on Fréchet regression. We establish Wasserstein error bounds that connect accuracy of the generated object distributions to the placement of landmarks, flow approximation and reconstruction error and prove consistency under finite second moments. This leads to convergence rates that depend on the metric complexity of the target objects, tail moments and fitting accuracy. Experiments show competitive accuracy with established flow matching methods in Euclidean simulations and lower Wasserstein errors than specifically tailored methods for several scientific datasets. Across simulation settings, we find that generation error decreases as sample size increases.
查看此版本解读 →查看此版本原文 ↗