Flow Matching is Adaptive to Manifold Structures

Shivam Kumar; Yixin Wang; Lizhen Lin

stat.MLFeb 25, 2026

Flow Matching is Adaptive to Manifold Structures

Authors: Shivam Kumar, Yixin Wang, Lizhen Lin

Organizations: Booth School of Business, University of Chicago · Department of Statistics, University of Michigan · Department of Mathematics, University of Maryland, College Park

Abstract

Flow matching has emerged as a simulation-free alternative to diffusion-based generative modeling, producing samples by solving an ODE whose time-dependent velocity field is learned along an interpolation between a simple source distribution (e.g., a standard normal) and a target data distribution. Flow-based methods often exhibit greater training stability and have achieved strong empirical performance in high-dimensional settings where data concentrate near a low-dimensional manifold, such as text-to-image synthesis, video generation, and molecular structure generation. Despite this success, existing theoretical analyses of flow matching assume target distributions with smooth, full-dimensional densities, leaving its effectiveness in manifold-supported settings largely unexplained. To this end, we theoretically analyze flow matching with linear interpolation when the target distribution is supported on a smooth manifold. We establish a non-asymptotic convergence guarantee for the learned velocity field, and then propagate this estimation error through the ODE to obtain statistical consistency of the implicit density estimator induced by the flow-matching objective. The resulting convergence rate is near minimax-optimal, depends only on the intrinsic dimension, and reflects the smoothness of both the manifold and the target distribution. Together, these results provide a principled explanation for how flow matching adapts to intrinsic data geometry and circumvents the curse of dimensionality.

Figures & tables

	Key assumptions	Low-dimensional structure	Velocity field estimation	Optimality	Metric
Albergo and Vanden-Eijnden (2022)	$\mathbf{x}\mapsto\widehat{v}(\mathbf{x},t)$ is $\widehat{K}$ -Lipschitz	✗	✗	✗	${\mathrm{W}}_{2}$
Fukumizu et al. (2024)	Bounded support $\mathbf{x}\mapsto{v^{\star}}(\mathbf{x},t)$ is differentiable with $\\|\nabla_{\mathbf{x}}v^{\star}(\mathbf{x},t)\\|_{\text{op}}\lesssim\tfrac{1}{1-t}$	✗	✓	✓	${\mathrm{W}}_{2}$
Gao et al. (2024a)	Log-concave and Gaussian mixture targets $\mathbf{x}\mapsto{v^{\star}}(\mathbf{x},t)$ is $L_{t}$ -Lipschitz	✗	✓	✗	${\mathrm{W}}_{2}$
Zhou and Liu (2025)	Bounded support Lipschitz score function	✗	✓	✗	${\mathrm{W}}_{2}$
Kunkel and Trabs (2025)	Bounded support $\mathbf{x}\mapsto{v^{\star}}(\mathbf{x},t)$ is Lipschitz continuous	✓ single-chart manifold projected ${\mathrm{W}}_{1}$	✓ (exponential size network)	✓	${\mathrm{W}}_{1}$
Ours	Bounded support ${\mu_{2}\!\left({\partial_{\mathbf{x}}v^{\star}}(\mathbf{x},t)\right)}\lesssim\tfrac{1}{(1-t)^{1-\xi}}$ $\xi\approx(\log\log(n))^{-1}$	✓ (general manifold)	✓	✓	${\mathrm{W}}_{2}$

Table 1: Comparison with existing theoretical results for flow matching.

$\mathsf{d}$	$\mathsf{D}$	${\mathrm{W}}_{1,\mathrm{slice}}^{\mathrm{std}}$	$\mathrm{dist}_{\mathcal{M}}$
	4	0.04177 $\pm$ 0.01935	0.05304 $\pm$ 0.00460
2	6	0.03788 $\pm$ 0.00725	0.05920 $\pm$ 0.00339
	8	0.04194 $\pm$ 0.01330	0.05861 $\pm$ 0.00589
	6	0.03277 $\pm$ 0.00573	0.07028 $\pm$ 0.00140
3	9	0.03994 $\pm$ 0.00997	0.06962 $\pm$ 0.00622
	12	0.04648 $\pm$ 0.01795	0.07906 $\pm$ 0.00353

Table 2: Mean and standard deviation of

{\mathrm{W}}_{1,\mathrm{slice}}^{\mathrm{std}}

and

\mathrm{dist}_{\mathcal{M}}

for estimated density in Example 1 across

(\mathsf{d},\mathsf{D})

$\mathsf{d}$	$\mathsf{D}$	${\mathrm{W}}_{1,\mathrm{slice}}^{\mathrm{std}}$	$\mathrm{dist}_{\mathcal{M}}$
	4	0.04407 $\pm$ 0.01421	0.05022 $\pm$ 0.00291
2	6	0.02430 $\pm$ 0.00407	0.06331 $\pm$ 0.00212
	8	0.03548 $\pm$ 0.01123	0.07248 $\pm$ 0.00218
	6	0.02998 $\pm$ 0.01201	0.07268 $\pm$ 0.00218
3	9	0.03790 $\pm$ 0.01672	0.09524 $\pm$ 0.00215
	12	0.03792 $\pm$ 0.00951	0.11211 $\pm$ 0.00311

Table 3: Mean and standard deviation of

{\mathrm{W}}_{1,\mathrm{slice}}^{\mathrm{std}}

and

\mathrm{dist}_{\mathcal{M}}

for estimated density in Example 2 across

(\mathsf{d},\mathsf{D})

Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 1: Comparison of generated samples and training data for Example 3 . The learned flow generates samples that recover the petal geometry and place negligible mass in the regions between segments.

Panel of Figure 2 — Figure 2: Real (left) vs. generated (right) MNIST samples.

Digit	$\mathrm{W}_{1,\mathrm{slice}}^{784}$	$\mathrm{W}_{1,\mathrm{slice}}^{\mathrm{BL}}$
0	$0.0229$	$0.0212$
1	$0.0158$	$0.0113$
2	$0.0273$	$0.0195$
3	$0.0265$	$0.0183$
4	$0.0246$	$0.0177$
5	$0.0283$	$0.0218$

Appendix

Table 4: Per-digit evaluation on MNIST (

n_{\mathrm{eval}}=1000

$n$	$\mathrm{W}_{1,\mathrm{slice}}^{784}$
100	$0.0416$
250	$0.0438$
500	$0.0406$
1000	$0.0325$
2000	$0.0275$
5000	$0.0254$

Appendix

Table 5: Sample complexity ablation for digit 3.

Figure 3: Log-log regression for digit 3. Points show empirical

\mathrm{W}_{1,\mathrm{slice}}^{784}

; solid line shows the power-law fit

\mathrm{W}_{1,\mathrm{slice}}^{784}\propto n^{-0.152}

; dashed horizontal line indicates baseline

\mathrm{W}_{1,\mathrm{slice}}^{\mathrm{BL}}=0.0180

	$\mathsf{d}=2$			$\mathsf{d}=3$		$\mathsf{d}=4$
$n$	$\mathsf{D}=6$	$\mathsf{D}=9$	$\mathsf{D}=12$	$\mathsf{D}=8$	$\mathsf{D}=12$	$\mathsf{D}=10$
256	$0.043\pm 0.018$	$0.052\pm 0.018$	$0.045\pm 0.014$	$0.047\pm 0.013$	$0.057\pm 0.011$	$0.042\pm 0.006$
512	$0.029\pm 0.008$	$0.043\pm 0.017$	$0.035\pm 0.009$	$0.034\pm 0.009$	$0.036\pm 0.005$	$0.028\pm 0.007$
1024	$0.033\pm 0.006$	$0.027\pm 0.008$	$0.034\pm 0.005$	$0.029\pm 0.005$	$0.033\pm 0.005$	$0.025\pm 0.004$
2048	$0.022\pm 0.005$	$0.024\pm 0.004$	$0.026\pm 0.005$	$0.020\pm 0.005$	$0.022\pm 0.006$	$0.023\pm 0.003$
4096	$0.019\pm 0.004$	$0.019\pm 0.005$	$0.017\pm 0.005$	$0.017\pm 0.003$	$0.020\pm 0.005$	$0.019\pm 0.005$

Appendix

Table 6: Sample complexity ablation on the sphere

\mathbbm{S}^{\mathsf{d}}\subset\mathbbm{R}^{\mathsf{D}}

Figure 4: Sample complexity on the sphere (log-log). Solid: empirical

\mathrm{W}_{1,\mathrm{slice}}^{\mathrm{std}}

; dashed: power-law fit; dotted: baseline

\mathrm{W}_{1,\mathrm{slice}}^{\mathrm{std,BL}}

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Oct 2, 2026J/K move · Enter open · S save

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

Open paper AskPDF arXiv

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

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

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