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A theoretical and empirical comparison claiming diffusion bridges have lower stochastic-optimal-control cost and greater robustness than flow matching when training data are scarce.

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2026-08-04 13:52 UTC pith:J5VZQJ7C

arxiv 2509.24531 v2 pith:J5VZQJ7C submitted 2025-09-29 cs.CV

Diffusion Bridge or Flow Matching? A Unifying Framework and Comparative Analysis

Kaizhen Zhu , Mokai Pan , Zhechuan Yu , Jingya Wang , Jingyi Yu , Ye Shi This is my paper
classification cs.CV
keywords bridgediffusionflowmatchingtheoreticaldataempiricalexperiments
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper asks a practical question: when you want a model that turns one image distribution into another, for example low-quality to high-quality or a photo to a painting, should you use a diffusion bridge or flow matching? The authors build the same transformer backbone for both methods so the comparison is fair, then test them on inpainting, super-resolution, deblurring, denoising, style transfer, and translation, with different training set sizes.

They claim two theoretical results. First, using stochastic optimal control, they say the bridge's control cost is always no larger than flow matching's cost, because the bridge adds a drift term that pushes the trajectory toward the target. The inequality itself is a simple consequence of their setup, but the paper goes further and says lower control cost means more stable and natural images, which is an interpretation, not a proof. Second, they claim that flow matching's linear interpolation between paired samples, the t and 1-t schedule, becomes invalid when the training data are limited, because the empirical measures are discrete and do not satisfy the smoothness condition of Brenier's optimal transport theorem. That step is the weakest part: flow matching trains on sample pairs, and linear interpolation of pairs remains a well-defined path even for discrete data. The empirical measures being atomic does not by itself predict that flow matching should collapse at small sample sizes, so the theoretical story does not actually explain the experimental pattern.

The experiments themselves show a consistent pattern: the diffusion bridge has better perceptual scores, especially when data are scarce or the transformation is large, while flow matching trains and samples faster. That pattern is useful evidence, but it needs

Core claim

The paper's load-bearing assertion is Theorem 4.2: J(u*DB) <= J(u*FM), interpreted as DB producing 'more stable and natural trajectories,' and Remark 4.3: for finite empirical measures, the interpolation coefficients t and 1-t of FM 'lose their validity' and the path is no longer an optimal transport interpolation. If correct, DB would be the preferable method for small training data and large distributional discrepancy.

Load-bearing premise

The Section 4.2 argument assumes that the validity of FM's linear schedule is governed by Brenier's absolute-continuity condition. The paper identifies the training measures as atomic and concludes that McCann's interpolation is undefined, therefore the t,1-t coefficients become ineffective. This conflates the well-definedness of sample-wise linear interpolation (what FM actually trains on) with existence of a continuous Brenier map; a discrete optimal transport plan between paired samples still defines a valid Wasserstein geodesic. Without this premise, the theoretical prediction of FM collapse at small n has no support. Location: Section 4.2, Remark 4.3 and Appendix A.3.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Axiom & Free-Parameter Ledger

4 free parameters · 6 axioms · 0 invented entities

The central cost inequality depends on the closed-form DB controller imported from UniDB (Zhu et al., 2025, same authors), and the data-size prediction depends on a controversial reading of Brenier's theorem applied to empirical measures. The practical experiments use several chosen hyperparameters (lambda^2, schedule s, terminal margin, network dimensions) that are not tied to the theory.

free parameters (4)
  • lambda^2 (steady variance level) = 30^2/255^2
    Chosen in Appendix D from prior bridge work; scales the noise schedule and is a hyperparameter, though the cost inequality in Theorem 4.2 holds for any positive lambda^2.
  • terminal margin e^{-theta_bar_T} = 0.005
    Chosen in Appendix D to avoid exact terminal hitting; affects the practical DB sampling but not the proof.
  • cosine schedule parameter s = 0.008
    Flipped cosine noise schedule for theta_t; taken from prior bridge papers and adjusted by hand.
  • network and training hyperparameters = patch=2, hidden=1024, depth=24, heads=16, MLP ratio=4.0, batch=8, steps=600k, LR=1e-4
    Selected by hand for the DiT architecture; not tied to the theoretical claims but needed for the experiments.
axioms (6)
  • standard math Brenier theorem and McCann interpolation for absolutely continuous measures
    Invoked in Section 4.2 to characterize the optimal transport map and the dynamic OT solution.
  • domain assumption Certainty equivalence principle reduces the SDE SOC problem to an ODE SOC problem
    Used in Section 4.1 to make the DB and FM SOC formulations comparable; cited from prior work.
  • domain assumption The closed-form DB optimal controller from UniDB is correct
    Equation (17) takes u*DB from Zhu et al. 2025, a paper by the same authors; Theorem 4.2 depends on it.
  • ad hoc to paper Lower control cost J implies more stable and natural trajectories and better perceptual quality
    Stated after Theorem 4.2 and used to connect the cost inequality to image quality; no argument is given.
  • ad hoc to paper Absence of absolute continuity for empirical measures implies the t,1-t interpolation coefficients lose validity
    Remark 4.3 makes this inference, but the paper does not derive it from the failure of McCann interpolation.
  • domain assumption All finite-pixel images lie in a compact space and a relevant class is absolutely continuous
    Assumed in Section 4.2 before invoking Brenier; real images are discrete-valued, so the assumption is inconsistent with the actual data.

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Diffusion Bridge and Flow Matching have both demonstrated compelling empirical performance in transformation between arbitrary distributions. However, there remains confusion about which approach is generally preferable, and the substantial discrepancies in their modeling assumptions and practical implementations have hindered a unified theoretical account of their relative merits. We have, for the first time, provided a unified theoretical and experimental validation of these two models. We recast their frameworks through the lens of Stochastic Optimal Control and prove that the cost function of the Diffusion Bridge is lower, guiding the system toward more stable and natural trajectories. Simultaneously, from the perspective of Optimal Transport, interpolation coefficients $t$ and $1-t$ of Flow Matching become increasingly ineffective when the training data size is reduced. To corroborate these theoretical claims, we propose a novel, powerful architecture for Diffusion Bridge built on a latent Transformer, and implement a Flow Matching model with the same structure to enable a fair performance comparison in various experiments. Comprehensive experiments are conducted across Image Restoration, Translation, and Style Transfer tasks, systematically varying both the distributional discrepancy (different difficulty) and the training data size. Extensive empirical results align perfectly with our theoretical predictions and allow us to delineate the respective advantages and disadvantages of these two models. Our code is available at https://github.com/zhukaizhen/diffusion_bridge_flow_matching.

Figures

Figures reproduced from arXiv: 2509.24531 by Jingya Wang, Jingyi Yu, Kaizhen Zhu, Mokai Pan, Ye Shi, Zhechuan Yu.

Figure 1
Figure 1. Figure 1: Overview of Diffusion Bridge (DB) versus Flow Matching (FM) on mapping one distri [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Qualitative comparison of visual results between Diffusion Bridge and Flow Matching on [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The performance (LPIPS and FID) of Flow Matching and Diffusion Bridge under Image [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Qualitative comparison of visual results between Diffusion Bridge and Flow Matching [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The performance (FID) of Flow Matching and Diffusion Bridge under Image Inpainting [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Training (with training data size 27000) and inference time (NFEs = 20) comparison on [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Qualitative comparison of visual results between Diffusion Bridge and Flow Matching in [PITH_FULL_IMAGE:figures/full_fig_p023_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Additional Diffusion Bridge visual results on Image Translation with CelebFaces dataset. [PITH_FULL_IMAGE:figures/full_fig_p023_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Additional Diffusion Bridge visual results on Style-Transfer with CelebA-HQ dataset. [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Additional visual comparison between Diffusion Bridge and Flow Matching on 4 [PITH_FULL_IMAGE:figures/full_fig_p024_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Additional Diffusion Bridge and Flow Matching visual results on Image Inpainting [PITH_FULL_IMAGE:figures/full_fig_p025_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Additional Flow Matching visual results on Image Inpainting (different masks and different [PITH_FULL_IMAGE:figures/full_fig_p026_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Additional Flow Matching visual results on Image Inpainting (different masks and different [PITH_FULL_IMAGE:figures/full_fig_p027_13.png] view at source ↗

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Cited by 1 Pith paper

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    RDBM reformulates generalized diffusion bridge SDEs to use distribution residuals for adaptive noise modulation, unifying prior bridge models as special cases and achieving SOTA on image restoration tasks.

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