Generalization Properties of Score-matching Diffusion Models for Intrinsically Low-dimensional Data

2026-10-01

AI Summary

This study addresses the limitations of existing generalization analyses for flow matching models, which typically rely on stringent assumptions and fail to characterize the intrinsic low-dimensional structure of data. By leveraging the Wasserstein-$p$ distance and finite-sample theory in conjunction with neural network architecture and hyperparameter optimization, this work investigates the statistical generalization of flow matching models in learning unknown distributions and derives finite-sample error bounds. It is demonstrated that the convergence rate depends solely on the intrinsic dimensionality of the data rather than the ambient dimension, effectively mitigating the curse of dimensionality under milder assumptions. Furthermore, high-probability upper error bounds are established, providing a rigorous theoretical explanation for the empirical success of flow matching models on structured data.

Abstract

Despite the remarkable empirical success of flow-matching models, their statistical generalization guarantees remain underdeveloped. Existing analyses often impose restrictive assumptions on the estimated velocity field and yield convergence rates that fail to reflect the intrinsic low-dimensional structure common in real data, such as natural images and molecular geometries. In this work, we study the statistical generalization of flow-matching models for learning an unknown distribution $P_{\mathrm{data}}$ from finitely many samples. We derive finite-sample error bounds on the learned generative distribution, measured in the Wasserstein-$p$ distance, for all $p\geq 1$. Specifically, given $n$ i.i.d. samples from $P_{\mathrm{data}}$, we show that, for every $d>d_p^\ast(P_{\mathrm{data}})$ and appropriately chosen network architectures and hyperparameters, the learned distribution $\widehat{P}^{\mathrm{FM}}$ satisfies $ \mathbb{W}_p(\widehat{P}^{\mathrm{FM}},P_{\mathrm{data}}) \lesssim n^{-1/d}+n^{-1/(2p)}\bigl(\log(1/ξ)\bigr)^{1/(2p)}$ with probability at least $1-ξ$, 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 data and mitigates the curse of dimensionality, as the convergence exponent depends on the intrinsic rather than ambient dimension. These guarantees remain meaningful in high-dimensional regimes and provide a theoretical explanation for the empirical success of flow matching on structured data distributions under substantially more relaxed assumptions than those in existing analyses.

Problem

• Flow-matching models
• Statistical generalization
• Intrinsic low-dimensional data
• Wasserstein distance
• Curse of dimensionality

Innovation

• Flow-matching models
• Statistical generalization
• Wasserstein distance
• Intrinsic dimensionality
• Curse of dimensionality