Guidance Without Labels: Flow Matching from Coupled Gaussian Mixtures

Stanislas Ducotterd; Bassam El Rawas; Michael Unser

cs.LGOct 12, 2026

Guidance Without Labels: Flow Matching from Coupled Gaussian Mixtures

Authors: Stanislas Ducotterd, Bassam El Rawas, Michael Unser

Organizations: Biomedical Imaging Group, EPFL Lausanne, Switzerland

Abstract

Flow matching models are typically trained to transport a Gaussian source to the data distribution, with source and target samples paired independently. We replace the source with a fitted Gaussian mixture and couple each data point to a mixture component through an entropic optimal transport problem. We show that the resulting transport reduces to the standard Gaussian-source problem up to a guidance-like correction, which enables the reuse of a pretrained velocity field. This correction is a closed-form function of the fitted mixture plus a term that we learn with a small correction network. We test our model on published checkpoints for CIFAR-10 and CelebA-HQ. It improves FID at every step count, and on CelebA-HQ it reaches the quality of the baseline with about half the steps. The closed-form term alone, with no training, already delivers most of this improvement; the small correction network accounts for the rest.

Figures & tables

Figure 1: Flow matching toward the checkerboard density with different sources and couplings, one model per row. Left: marginals at intermediate times $t$ along a $256$ -step Euler integration. Right: the final distribution at $t=1$ when sampling with a small number of Euler steps.

$K$	64	256	1024	$\mathbf{4096}$	16384
train	$-0.412$	$-0.358$	$-0.305$	$-0.259$	$-0.178$
test	$-0.412$	$-0.366$	$-0.331$	$\mathbf{-0.319}$	$-0.330$

Table 1: CIFAR-10 log-likelihood per dimension of the fitted mixture, on train and test set.

CelebA-HQ $256$ , FID $\downarrow$
Euler steps	1	2	4	8	16	32	64	128	256
Gaussian	258.40	182.43	137.16	89.01	48.53	25.08	15.00	11.45	10.01
GMM untrained	186.94	123.86	93.61	63.60	37.12	19.91	12.81	10.63	9.63
GMM trained	182.97	115.04	87.38	58.90	33.40	16.69	11.24	9.80	9.42
CIFAR-10, FID $\downarrow$
Gaussian	371.25	173.07	52.95	17.20	8.59	5.47	4.13	3.43	3.05

Table 2: FID and Inception Score across Euler step counts,

10

k samples on CelebA-HQ

256

(

K=1024

) and

50

k on CIFAR-10 (

K=4096

), against the full training sets.

Figure 2: The mean of a single component (left) and seven samples of the coupled model conditioned on it, $256$ Euler steps, on CelebA-HQ. The CIFAR-10 counterpart is in Appendix B.3 .

CelebA-HQ, pixel $L^{2}$
	mean	std	min	$p_{1}$	$p_{5}$	median	max
GMM coupled	162.0	22.9	95.8	110.9	125.5	161.8	231.4
Gaussian	159.7	22.7	95.3	111.2	123.9	159.2	241.3
CIFAR-10, pixel $L^{2}$
GMM coupled	18.35	3.98	6.00	9.12	11.66	18.51	30.92
Gaussian	18.10	4.24	6.69	8.39	11.24	18.15	30.09

Table 3: Distance of

1000

samples per model to their nearest training image, in pixel space.

Figure 3: The four generated samples closest to the training set, out of $1000$ , in pixel space, each next to its nearest training image, on CelebA-HQ. The CIFAR-10 counterpart is in Appendix B.3 .

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Euler steps	1	2	4	8	16	32	64	128	256
Gaussian	258.40	182.43	137.16	89.01	48.53	25.08	15.00	11.45	10.01
GMM untrained, $K=4$	252.49	174.30	122.69	83.35	48.33	24.38	14.24	10.99	9.80
GMM untrained, $K=16$	242.64	163.23	111.51	75.50	44.80	22.94	13.97	10.54	9.77
GMM untrained, $K=64$	233.41	153.49	106.58	71.06	41.19	21.54	13.59	10.60	9.66
GMM untrained, $K=256$	218.41	141.49	99.99	66.99	38.61	20.94	13.01	10.42	9.73
GMM untrained, $K=1024$	186.94	123.86	93.61	63.60	37.12	19.91	12.81	10.63	9.63

Appendix

Table 4: CelebA-HQ

256

, FID

\downarrow

of the untrained model (

{\bf{W}}_{\phi}=\mathbf{0}

) for every mixture size

K

10

k samples against the full training set, uniform Euler grid in

s(t)

. The

K=1024

row is the one of Table 2 .

Euler steps	1	2	4	8	16	32	64	128	256
Gaussian	371.25	173.07	52.95	17.20	8.59	5.47	4.13	3.43	3.05
GMM untrained, $K=4$	415.10	142.93	43.32	15.61	8.27	5.50	4.01	3.34	3.02
GMM untrained, $K=16$	357.96	130.65	39.43	15.12	8.40	5.51	4.04	3.40	3.03
GMM untrained, $K=64$	312.40	120.00	36.35	14.81	8.39	5.57	4.02	3.38	2.98
GMM untrained, $K=256$	250.91	108.81	34.59	14.89	8.47	5.54	4.05	3.38	2.97
GMM untrained, $K=1024$	197.89	95.62	32.97	14.59	8.49	5.58	4.05	3.34	2.95

Appendix

Table 5: CIFAR-10, FID

\downarrow

of the untrained model (

{\bf{W}}_{\phi}=\mathbf{0}

) for every mixture size

K

50

k samples against the full training set, uniform Euler grid in

s(t)

. The

K=4096

row is the one of Table 2 .

CelebA-HQ, latent $L^{2}$
	mean	std	min	$p_{1}$	$p_{5}$	median	max
GMM coupled	9.18	1.95	5.12	5.92	6.56	8.87	17.23
Gaussian	9.11	1.94	5.30	5.83	6.43	8.83	18.09
CIFAR-10, latent $L^{2}$
GMM coupled	12.09	2.51	5.59	6.48	7.91	12.33	18.24
Gaussian	11.99	2.53	5.56	6.31	7.86	12.22	18.18

Appendix

Table 6: Distance of

1000

samples per model to their nearest training image in the Inception latent space. The CIFAR-10 test-set rows are a reference of novel images.

Figure 4: CIFAR-10: the mean of a single component (left) and seven samples of the coupled model conditioned on it, $256$ Euler steps.

Figure 5: CIFAR-10: the four generated samples closest to the flip-symmetrized training set, out of $1000$ , by pixel $L^{2}$ , each next to its nearest training image.

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Flow Matching trains continuous-time generative models by regressing the velocity field of a probability path between a simple source distribution and a target data distribution. The coupling that pairs source and target samples strongly affects optimization and sample quality, but structured couplings typically rely on mini-batch transport or assignment procedures whose cost grows at least quadratically in batch size. We propose Quantile Coupling Flow Matching (QC-FM), a lightweight one-sided coupling: rather than matching two pre-sampled batches, it samples only the data batch and constructs each paired source directly. Data ranks projected along a small number of random orthogonal directions are mapped to Gaussian quantiles, and the latent code is completed in the orthogonal complement by conditional Gaussian sampling. The construction is one-dimensional per slice, so the coupling requires no pairwise cost matrix and no assignment to solve. We show that, for each drawn frame, this coupling eliminates the irreducible regression variance along every selected slice and makes the ideal flow exactly straight there, while leaving the sampling prior unchanged: generation still starts from the standard Gaussian, and the training source deviates from it only through the copula of the slice codes, whose transport cost we bound. For training, we apply QC to an anchor subset and complete the remaining source slots with exact Gaussian samples, retaining the QC bias while preserving an explicit signal from the Baseline coupling. Across CIFAR-10, CelebA, FFHQ, and ImageNet-64, QC-FM improves over the Baseline under matched training budgets, reducing FID by up to 12.9%, and outperforms OT-CFM on all four datasets. These results suggest that preserving projected rank structure is a simple and scalable way to inject useful geometric bias into FM couplings without solving a mini-batch transport problem.

Abstract

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