将流匹配推广到李群，用指数曲线替代直线段，实现高效生成建模。
Flow Matching on Lie Groups
- 用李群的指数曲线替代欧氏空间的直线，构建内在的流匹配方法
- 在矩阵李群上可直接用矩阵运算快速实现，计算高效
- 适合处理含姿态与特征的联合数据，如等变神经场的潜在编码
流匹配（Flow Matching, FM）是一种新兴的生成建模技术：通过将容易采样的分布 $\mathfrak{X}_0$ 中的样本流动至目标分布 $\mathfrak{X}_1$ 来实现采样。关键在于，在给定终点的前提下训练流场，沿直线段移动至终点（Lipman et al. 2022）。然而直线仅在欧氏空间中定义良好。因此，Chen 和 Lipman（2023）将方法推广至黎曼流形，用测地线或其谱近似替代直线段。本文提出另一种视角：将流匹配推广至李群，以指数曲线替代直线段。该方法在多个矩阵李群上实现简单、内在且快速，因李群运算（乘积、逆、指数、对数）即对应矩阵运算。该方法可用于包含特征（$\mathbb{R}^n$）和姿态（某李群）的数据生成建模，例如等变神经场（Equivariant Neural Fields, Wessels et al. 2025）的潜在编码。
原文摘要 · Abstract (English)
Flow Matching (FM) is a recent generative modelling technique: we aim to learn how to sample from distribution $\mathfrak{X}_1$ by flowing samples from some distribution $\mathfrak{X}_0$ that is easy to sample from. The key trick is that this flow field can be trained while conditioning on the end point in $\mathfrak{X}_1$: given an end point, simply move along a straight line segment to the end point (Lipman et al. 2022). However, straight line segments are only well-defined on Euclidean space. Consequently, Chen and Lipman (2023) generalised the method to FM on Riemannian manifolds, replacing line segments with geodesics or their spectral approximations. We take an alternative point of view: we generalise to FM on Lie groups by instead substituting exponential curves for line segments. This leads to a simple, intrinsic, and fast implementation for many matrix Lie groups, since the required Lie group operations (products, inverses, exponentials, logarithms) are simply given by the corresponding matrix operations. FM on Lie groups could then be used for generative modelling with data consisting of sets of features (in $\mathbb{R}^n$) and poses (in some Lie group), e.g. the latent codes of Equivariant Neural Fields (Wessels et al. 2025).
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