  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.
