Linear Convergence of Diffusion Models Under the Manifold Hypothesis

Score-matching generative models have proven successful at sampling from complex high-dimensional data distributions. In many applications, this distribution is believed to concentrate on a much lower $d$-dimensional manifold embedded into $D$-dimensional space; this is known as the manifold hypothesis. The current best-known convergence guarantees are either linear in $D$ or polynomial (superlinear) in $d$. The latter exploits a novel integration scheme for the backward SDE. We take the best of both worlds and show that the number of steps diffusion models require in order to converge in Kullback-Leibler~(KL) divergence is linear (up to logarithmic terms) in the intrinsic dimension $d$. Moreover, we show that this linear dependency is sharp.

Metric Flow Matching for Smooth Interpolations on the Data Manifold

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.