Euclidean Distance-Based Flow Matching

Paul-Henrik Heilmann; Peter Ochs

cs.LGOct 13, 2026

Euclidean Distance-Based Flow Matching

Authors: Paul-Henrik Heilmann, Peter Ochs

Organizations: Saarland University, Saarbrücken, Germany

Abstract

Flow Matching has recently become popular as a technique for generative modeling with applications in many domains. In the standard Flow Matching formulation, we interpret the vector field as the solution of a least-squares regression problem, which is known to be sensitive to contamination. Therefore we leverage the viewpoint of robust statistics and replace the squared Euclidean distance with the Euclidean distance as a regression objective. This seemingly minor change strongly affects the properties of Flow Matching. We provide a probabilistic interpretation of the new regression target as a conditional geometric median and use this characterization to study distributional robustness of the Euclidean Distance-based Flow Matching. We analyze the mathematical properties of this changed objective and the effect of a contaminated target distribution on the involved conditional velocity distributions. We discuss the challenge of directly characterizing the distribution that our method induces. In specific contamination settings we show theoretically that our method yields an improvement for severe contamination. Finally, we present numerical experiments conducted on the CIFAR-10 image dataset where a proportion of the samples has been contaminated. Our Euclidean distance-based Flow Matching shows a clear advantage in scenarios of severe contamination of the target distribution.

Figures & tables

Contamination	Description	Parameters
Random noise	Replace the entire image by independent uniform noise	$U[-1,1]$
Gaussian noise	Add independent Gaussian noise and clip to $[-1,1]$	$\sigma=0.15$
Salt and pepper	Replace selected pixels by either $-1$ or $1$	$\rho=0.05$
Patch noise	Replace a uniformly located square patch by uniform noise	$8\times 8$ patch
Inversion	Invert all normalized pixel values, $T(x)=-x$	–
Gaussian blur	Convolve the image with an isotropic Gaussian kernel	$\sigma=1.5$ , $11\times 11$

Table 1: Contamination mechanisms used in the experiments. Images represented in

[-1,1]^{3\times 32\times 32}

Objective	FID
Squared	$3.70\pm 0.08$
Unsquared	$\mathbf{3.50}\pm 0.01$

Table 2: FID on clean CIFAR-10 (lower is better). Entries report the mean

\pm

standard deviation over three seeds. The lower mean is shown in bold.

Contamination	Objective	$\epsilon=0.05$	$\epsilon=0.10$	$\epsilon=0.15$
Random noise	Squared	$9.69\pm 0.20$	$19.16\pm 0.33$	$29.73\pm 0.61$
	Unsquared	$\mathbf{6.82}\pm 0.26$	$\mathbf{12.21}\pm 0.28$	$\mathbf{19.40}\pm 0.82$
Gaussian noise	Squared	$3.79\pm 0.06$	$4.92\pm 0.17$	$6.72\pm 0.20$
	Unsquared	$\mathbf{3.51}\pm 0.07$	$\mathbf{4.17}\pm 0.06$	$\mathbf{5.31}\pm 0.05$
Salt and pepper	Squared	$4.45\pm 0.04$	$6.49\pm 0.17$	$9.44\pm 0.53$
	Unsquared	$\mathbf{3.92}\pm 0.10$	$\mathbf{5.00}\pm 0.09$	$\mathbf{6.65}\pm 0.32$

Table 3: FID on clean CIFAR-10 (lower is better) after training under contamination. Entries report the mean

\pm

standard deviation over three seeds. The lower mean within each pair is bold.

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 1 — Figure 1: Representative examples of the seven contamination mechanisms considered in our experiments. Except for random-noise replacement, all examples are obtained from the same clean CIFAR-10 image using the parameters reported in Table 1 .

Panel of Figure 2 — Figure 2: Generated samples for Random noise with contamination level $\epsilon=0.15$ . The same source samples are used for squared and $L_{2,1}$ Flow Matching.

Panel of Figure 3 — Figure 3: Generated samples for Gaussian noise with contamination level $\epsilon=0.15$ . The same source samples are used for squared and $L_{2,1}$ Flow Matching.

Panel of Figure 4 — Figure 4: Generated samples for Salt-and-Pepper noise with contamination level $\epsilon=0.15$ . The same source samples are used for squared and $L_{2,1}$ Flow Matching.

Panel of Figure 5 — Figure 5: Generated samples for Patch noise with contamination level $\epsilon=0.15$ . The same source samples are used for squared and $L_{2,1}$ Flow Matching.

Panel of Figure 6 — Figure 6: Generated samples for Inversion with contamination level $\epsilon=0.15$ . The same source samples are used for squared and $L_{2,1}$ Flow Matching.

Panel of Figure 7 — Figure 7: Generated samples for Gaussian blur with contamination level $\epsilon=0.15$ . The same source samples are used for squared and $L_{2,1}$ Flow Matching.

Panel of Figure 8 — Figure 8: Generated samples for Motion blur with contamination level $\epsilon=0.15$ . The same source samples are used for squared and $L_{2,1}$ Flow Matching.

Flow Matching

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

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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.

Abstract

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