arXiv:2611.15220v1 Announce Type: cross
Abstract: 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 Fr\'echetFlow, 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 Fr\'echet means of reference objects, drawing on Fr\'echet 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.
Abstract: 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 Fr\'echetFlow, 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 Fr\'echet means of reference objects, drawing on Fr\'echet 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.
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