位置-尺度族上的概率测地线流匹配

中文摘要 AI 辅助

流匹配（FM）近年来因其概念简单和强大的实证性能而成为生成建模的一个有前景的框架。在FM中，样本沿着由神经网络参数化的向量场进行传输，从而诱导出一条概率路径，该路径从简单的噪声分布演化到目标数据分布，并由常微分方程（ODE）控制。然而，现有的FM方法主要依赖于高斯分布之间最优传输（OT）导出的概率路径，这可能对于捕捉具有非均匀结构（如重尾或鲜明对比）的复杂数据而言是次优的。在这项工作中，我们将FM推广到更广泛的位置-尺度族，以处理数据非均匀性，并引入一类新颖的概率路径，这些路径被定义为概率分布流形上的测地线。我们将这种方法命名为概率测地线流匹配，以区别于先前在输入空间中定义的测地线（黎曼）FM方法。我们认为基于欧几里得OT的路径在概率空间中不一定是最优的，并且可能限制建模灵活性。通过不同规模的合成基准和科学数据集，我们证明了所提出的方法能更有效地捕捉复杂分布，与最先进的几何动机生成模型相比，实现了改进或相当的性能。

英文摘要

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.

发表机构

• School of Mathematical & Statistical Sciences（数学与统计科学学院）
• Arizona State University（亚利桑那州立大学）

机构由 AI 辅助整理，请以论文原文为准。

补充信息

备注

25 pages, 10 figures