Metric Flow Matching for Smooth Interpolations on the Data Manifold
This addresses the issue of capturing underlying dynamics in tasks like trajectory inference for researchers in fields such as biology and robotics, though it is an incremental improvement over existing flow matching methods.
The paper tackles the problem of generating unrealistic straight interpolations in generative models by proposing Metric Flow Matching (MFM), a framework that uses approximate geodesics on a data-induced Riemannian metric to produce smoother interpolations on the data manifold, achieving state-of-the-art results on single-cell trajectory prediction.
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.