On the minimax optimality of Flow Matching through the connection to kernel density estimation

Lea Kunkel; Mathias Trabs

stat.MLApr 17, 2025

On the minimax optimality of Flow Matching through the connection to kernel density estimation

Authors: Lea Kunkel, Mathias Trabs

Organizations: Ruhr University Bochum · Karlsruhe Institute of Technology

Abstract

Flow Matching has recently gained attention in generative modeling as a simple and flexible alternative to diffusion models. While existing statistical guarantees adapt tools from the analysis of diffusion models, we take a different perspective by connecting Flow Matching to kernel density estimation. We first verify that the kernel density estimator matches the optimal rate of convergence in Wasserstein distance up to logarithmic factors, improving existing bounds for the Gaussian kernel. Based on this result, we prove that for sufficiently large networks, Flow Matching achieves the optimal rate up to logarithmic factors. If the target distribution lies on a lower-dimensional manifold, we show that the kernel density estimator profits from the smaller intrinsic dimension on a small tube around the manifold. The faster rate also applies to Flow Matching, providing a theoretical foundation for its empirical success in high-dimensional settings.

Figures & tables

Panel of Figure 1 — Figure 1: Example results of the Flow Matching (FM) estimator and the KDE. The generated samples are the blue dots, the training sample is represented by black dots on the red sine curve. The top row shows generated samples of the FM estimator after $40000$ runs.

Panel of Figure 2 — Figure 2: Comparison between the Flow Matching estimator and the KDE in terms of distance to the manifold and the largest gap of the projection estimator on the manifold.

Figure 3: Setting for the Taylor expansion.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 4 — Figure 4: Comparison between the Flow Matching estimator and the KDE in terms of distance to the manifold and the largest gap of the projection estimator on the manifold. Top row uses a SeLU network with three-hidden-layers of width $32$ and bottom row uses a three-hidden-layer SeLU net of width $8$ .

Flow Matching

Flow matching learns continuous-time vector fields that transport simple noise distributions into data for efficient generative modeling.

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

Saptarshi Chakraborty, Quentin Berthet, Peter L. Bartlett

Department of Statistics, University of Michigan · Google DeepMind · Department of Statistics, University of California, Berkeley+1

Flow Matching Flow-Based Generative Modeling Generalization Bounds

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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.

Abstract

Flow Matching

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