Matching objectives underpin the success of modern generative models and rely on constructing conditional paths that transform a source distribution into a target distribution. Despite being a fundamental building block, conditional paths have been designed principally under the assumption of Euclidean geometry, resulting in straight interpolations. However, this can be particularly restrictive for tasks such as trajectory inference, where straight paths might lie outside the data manifold, thus failing to capture the underlying dynamics giving rise to the observed marginals. In this paper, we propose Metric Flow Matching (MFM), a novel simulation-free framework for conditional flow matching where interpolants are approximate geodesics learned by minimizing the kinetic energy of a data-induced Riemannian metric. This way, the generative model matches vector fields on the data manifold, which corresponds to lower uncertainty and more meaningful interpolations. We prescribe general metrics to instantiate MFM, independent of the task, and test it on a suite of challenging problems including LiDAR navigation, unpaired image translation, and modeling cellular dynamics. We observe that MFM outperforms the Euclidean baselines, particularly achieving SOTA on single-cell trajectory prediction.

翻译：匹配目标是现代生成模型成功的基础，其依赖于构建将源分布转换为目标分布的条件路径。尽管作为基本构建模块，条件路径的设计主要基于欧几里得几何的假设，导致产生直线插值。然而，这对于诸如轨迹推断等任务可能尤为受限，因为直线路径可能位于数据流形之外，从而无法捕捉产生观测边缘分布的潜在动力学。本文提出度量流匹配（MFM），这是一种用于条件流匹配的新型无仿真框架，其中插值曲线是通过最小化数据诱导黎曼度量的动能而学习的近似测地线。通过这种方式，生成模型匹配数据流形上的向量场，这对应于更低的不确定性和更有意义的插值。我们规定了独立于任务的一般性度量来实例化MFM，并在包括激光雷达导航、非配对图像翻译和细胞动力学建模在内的一系列挑战性问题中测试了它。我们观察到MFM优于欧几里得基线，特别是在单细胞轨迹预测上达到了最先进水平。