Generalization Bounds for Flow-matching Generative Models for Intrinsically Low-dimensional Data

Despite its remarkable empirical performance, the theoretical properties of flow-matching in terms of generalization accuracy remain largely unexplored. The current state-of-the-art generalization analyses not only impose stringent assumptions on the estimated velocity field, but also scale poorly with the ambient dimension—precisely the high-dimensional setting where flow matching is most commonly applied. In particular, the data distributions targeted by flow matching models, such as natural images, molecular geometries, etc., are commonly hypothesized to concentrate near a low-dimensional structures embedded in the high-dimensional ambient data space. However, existing analyses fail to reflect this geometry in their derived convergence rates. In this paper, we bridge the gap between the theory and practice of flow matching by establishing generalization bounds on the Wasserstein-$p$ distance between the learned distribution and the target measure, for any $p\ge 1$. Specifically, given $n$ independent and identically distributed samples from a target distribution $P_{\mathrm{data}}$ and for every $d>d_p^{\ast }(P_{\mathrm{data}})$, we show that for appropriate network and hyperparameter choices, $W_p({\hat{P}}^{\text{FM}},P_{\text{data}})\lesssim n^{-1/d}+n^{-\frac{1}{2p}}{\left(\log (1/\xi )\right)}^{\frac{1}{2p}}$ with probability at least $1-\xi$, where $d_p^{\ast }(P_{\mathrm{data}})$ denotes the Wasserstein-$p$ dimension of the target measure. Our results demonstrate that flow matching naturally adapts to the intrinsic geometry of the data, effectively mitigating the curse of dimensionality: the convergence exponent depends only on the intrinsic dimension, $d_p^{\ast }(P_{\mathrm{data}})$, rather than the ambient dimension of the feature space, yielding convergence rates that remain meaningful in high-dimensional regimes and offering a principled theoretical explanation for the strong empirical performance of flow matching on structured data distributions with significantly relaxed assumptions, compared to the existing literature.