A Plug-in Interpretation of Conditioning in Score-Based Diffusion Models

Libo Chen; Souvik Ghosh; Teo Deveney; Chris Budd; Vinay P. Namboodiri

cs.CVAug 19, 2026

A Plug-in Interpretation of Conditioning in Score-Based Diffusion Models

Authors: Libo Chen, Souvik Ghosh, Teo Deveney, Chris Budd, Vinay P. Namboodiri

Organizations: University of Bath, Bath, UK · International Institute of Information Technology Hyderabad, India

Abstract

We propose a conditioning mechanism for diffusion models based on multi-speed joint diffusion of the target and the condition. The mechanism learns an unconditional joint score network and enforces conditioning at inference via a plug-in correction term. The plug-in term separates the conditioning contribution from the learned unconditional dynamics, offering a transparent view of how the condition steers generation of the target distribution. Building on this, we derive explicit conditional reverse-time SDEs and approximate probability-flow ODEs, enabling principled and directly comparable conditional samplers. To reduce the induced ODE--SDE discrepancy, we introduce a log-Fokker--Planck residual regularization that improves ODE sampling quality. Experiments on conditional image generation tasks demonstrate competitive performance and support the effectiveness of the plug-in conditioning view. Additional ODE--SDE comparison experiments show that the log-Fokker--Planck residual regularization improves deterministic ODE sampling.

Figures & tables

Figure 1: Inpainting on CelebA. For each example, we show the ground truth, the masked observation, and samples generated by our method from left to right.

Method	PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$	JFID $\downarrow$
CDE	25.12	0.870	0.042	18.06
CDiffE	23.07	0.844	0.057	19.25
CMDE	24.92	0.864	0.044	17.07
Ours	25.38	0.876	0.046	15.99

Table 1: Conditional generation results on CelebA inpainting. Lower LPIPS and JFID are better; higher PSNR and SSIM are better. All score-based methods use the same backbone and evaluation protocol.

Method	PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$	JFID $\downarrow$
CDE	23.80	0.650	0.114	15.77
CDiffE	23.83	0.656	0.139	20.20
CMDE	23.91	0.654	0.109	15.68
HCFLOW	24.95	0.702	0.107	19.55
Ours	25.12	0.672	0.098	14.56

Table 2: Conditional generation results on CelebA

8\times

super-resolution. Our method achieves the best PSNR, LPIPS, and JFID, while HCFLOW obtains the highest SSIM.

Figure 2: Super-resolution on CelebA. From left to right: ground-truth high-resolution images, low-resolution observations, and samples generated by our method.

Dataset	Method	Grad.	LPIPS $\downarrow$	PSNR $\uparrow$	JFID $\downarrow$
FFHQ	DPS [ 3 ]	Yes	0.2001	29.392	26.1210
FFHQ	DiffPIR [ 21 ]	No	0.2097	26.577	18.8032
FFHQ	Ours	No	0.1984	29.916	20.1224
ImageNet256	DPS [ 3 ]	Yes	0.3390	26.166	44.0984
ImageNet256	DiffPIR [ 21 ]	No	0.2259	25.505	17.3510
ImageNet256	Ours	No	0.2272	28.122	16.7309

Table 3: Noisy random inpainting with a pretrained diffusion prior. All methods use the same pretrained prior, masks, perturbations, test images, and sampling budget

(\mathrm{NFE}=1000)

. Metrics are computed using

100

samples. We report LPIPS, PSNR, and JFID as representative measures of perceptual similarity, reconstruction fidelity, and conditional consistency, respectively.

Figure 3: Qualitative comparison on ImageNet256 noisy random inpainting. From left to right: ground truth, noisy observation, DPS, DiffPIR, and Ours. Compared with DPS and DiffPIR, our method produces reconstructions that are more consistent with the observation while maintaining realistic image content.

Method	$W_{2}(\mathrm{ODE},\mathrm{SDE})\downarrow$	$W_{2}(\mathrm{ODE},\mathrm{GT})\downarrow$	$W_{2}(\mathrm{SDE},\mathrm{GT})\downarrow$
Baseline	$0.0329\pm 0.0009$	$0.0469\pm 0.0016$	$0.0368\pm 0.0013$
CLOSE ( $\alpha=0.1$ )	$\mathbf{0.0323\pm 0.0011}$	$\mathbf{0.0452\pm 0.0018}$	$\mathbf{0.0347\pm 0.0011}$

Table 4: MNIST conditional inpainting test results. Lower values are better. The checkpoint is selected by minimizing

W_{2}(\mathrm{ODE},\mathrm{SDE})

. Results are reported as mean

\pm

std over

8

splits with

N=2048

test conditions.

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

$\beta=\sigma_{Y}/\sigma_{X}$	0.02	0.05	0.1	0.3	1.0
Score error	0.020	0.050	0.099	0.287	0.706
End-to-end rel. $W_{2}$	0.008	0.020	0.045	0.138	0.414

Appendix

Table 5: Exact plug-in approximation error as a function of the condition diffusion speed ratio

\beta

. The empirical score error decays approximately as

O(\beta^{0.98})

$\beta$	PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$
0.05	24.02	0.912	0.061
0.10	25.38	0.876	0.046
0.20	25.04	0.854	0.058
0.30	24.36	0.802	0.067

Appendix

Table 6: Ablation on

\beta

(condition diffusion speed ratio). Higher is better for PSNR/SSIM, lower is better for LPIPS.

Method	NFE	Backprop	s/img	FID $\downarrow$	SSIM $\uparrow$	JFID $\downarrow$
DPS	1000	yes	245.1	84.47	0.694	44.10
DiffPIR	1000	no	128.0	38.22	0.753	17.35
Ours	1000	no	130.8	39.64	0.764	16.73

Appendix

Table 7: Runtime and distributional evaluation on ImageNet256 noisy inpainting using one RTX 3090 with batch size 1. All methods use the same 100 images and sampling budget (NFE = 1000).

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Probability flow ODEs in score-based and reflected diffusion models

Rama CONT

Mathematical Institute, University of Oxford

Diffusion Sampling Diffusion Models Score-Based Generative Modeling Stochastic Differential Equations

Open paper AskPDF arXiv

Probability-flow ordinary differential equations (PF-ODEs) are widely used as deterministic samplers for score-based diffusion models. Their usual justification is that the Fokker--Planck equation of a diffusion can be rewritten as a continuity equation driven by the score function of the forward diffusion. This identity does not, however, guarantee that the resulting velocity field generates a well-posed flow. We provide theoretical insights into the design of such deterministic samplers for generative models based on diffusions and reflected diffusions. We identify sufficient conditions for a regular Lagrangian PF-ODE flow to exist; reverse sampling and invertibility require two-sided divergence control. For learned scores, sampler stability is controlled by an unweighted velocity error, exposing a mismatch with density-weighted score matching and motivating architectural control of Jacobians, divergence, growth, and compression. Under the manifold hypothesis, positive-time regularization justifies an early-stopped PF-ODE while constants deteriorate near the data endpoint; an explicit sphere example shows that the exact deterministic flow becomes singular as the noise level vanishes even though the diffusion marginals remain well defined. These theoretical insights translate into concrete design principles for stable, invertible, and constraint-preserving diffusion samplers. We illustrate the practical relevance of these design principles using controlled numerical experiments.

Abstract

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