Probabilistic Geodesic Flow Matching on Location-Scale Families
Abstract
Flow matching (FM) has recently emerged as a promising framework for generative modeling due to its conceptual simplicity and strong empirical performance. In FM, samples are transported along a vector field parameterized by a neural network, inducing a probability path that evolves from a simple noise distribution to the target data distribution, governed by an ordinary differential equation (ODE). However, existing FM approaches predominantly rely on probability paths derived from optimal transport (OT) between Gaussian distributions, which may be suboptimal for capturing complex data with inhomogeneous structures such as heavy tail or sharp contrast. In this work, we generalize FM to the broader class of location-scale families for handling data inhomogeneity and introduce a novel class of probability paths defined as geodesics on the manifold of probability distributions. We name this approach probabilistic geodesic flow matching to distinguish it from prior geodesic (Riemannian) FM methods defined in input space. We argue that Euclidean OT-based paths are not necessarily optimal in probability space and may limit modeling flexibility. Through synthetic benchmarks and scientific datasets at different scales, we demonstrate that the proposed method more effectively captures complex distributions, leading to improved or comparable performance compared with SOTA geometry-motivated generative models.
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BibTeX
@misc{yu2026probabilistic,
title = {{Probabilistic Geodesic Flow Matching on Location-Scale Families}},
author = {Zeyuan Yu and Zhi Chang and Shiwei Lan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.34613},
eprint = {2609.34613},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34613},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Probabilistic Geodesic Flow Matching on Location-Scale Families
AU - Zeyuan Yu
AU - Zhi Chang
AU - Shiwei Lan
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.34613
UR - https://arxiv.org/abs/2609.34613
AB - Flow matching (FM) has recently emerged as a promising framework for generative modeling due to its conceptual simplicity and strong empirical performance. In FM, samples are transported along a vector field parameterized by a neural network, inducing a probability path that evolves from a simple noise distribution to the target data distribution, governed by an ordinary differential equation (ODE). However, existing FM approaches predominantly rely on probability paths derived from optimal transport (OT) between Gaussian distributions, which may be suboptimal for capturing complex data with inhomogeneous structures such as heavy tail or sharp contrast. In this work, we generalize FM to the broader class of location-scale families for handling data inhomogeneity and introduce a novel class of probability paths defined as geodesics on the manifold of probability distributions. We name this approach probabilistic geodesic flow matching to distinguish it from prior geodesic (Riemannian) FM methods defined in input space. We argue that Euclidean OT-based paths are not necessarily optimal in probability space and may limit modeling flexibility. Through synthetic benchmarks and scientific datasets at different scales, we demonstrate that the proposed method more effectively captures complex distributions, leading to improved or comparable performance compared with SOTA geometry-motivated generative models.
ER -