Gradient-guided diffusion samplers provide flexible priors for inverse problems and conditional generation, but strong guidance can move the sampling trajectory into regions where the learned score is poorly supported. Existing tangent-projection strategies limit first-order departure from an iso-density surface, yet discard potentially useful normal motion and overlook the second-order departure induced by tangent motion on a curved surface. We introduce curvature-adaptive tubular correction (CAT), a training-free plugin that regulates both effects within a shared, noise-dependent geometric budget. CAT decomposes the guidance gradient into normal and tangent components, charges normal displacement at first order and tangent displacement according to directional curvature, and obtains their jointly optimal magnitudes from a one-dimensional dual equation. Armijo backtracking calibrates the resulting finite step against the actual guidance objective, while matrix-free directional derivatives avoid constructing the full score Jacobian. We establish local guarantees for the tubular approximation, uniqueness of the correction, and sufficient objective decrease. Across seven inverse problems on FFHQ and ImageNet, CAT improves the evaluated pixel- and latent-space host samplers, with particularly consistent gains in perceptual metrics. It also improves black hole reconstruction on InverseBench and yields the lowest FID among the compared methods at every tested classifier-free guidance scale, while maintaining stable saturation and contrast. These results support curvature-aware tubular control as a reusable mechanism for stabilizing diffusion guidance.
Figures & tables
Figure 1: CAT within one reverse-diffusion step. The host update can leave the safe region, whereas CAT uses curvature-adaptive normal–tangent allocation to produce xt− inside it.
Task
Type
Method
FFHQ
ImageNet
PSNR ↑
SSIM ↑
LPIPS ↓
FID ↓
PSNR ↑
SSIM ↑
LPIPS ↓
FID ↓
Super-resolution 4×
Pixel
DAPS
28.07
0.749
0.2773
76.15
25.54
0.642
0.3759
113.16
DCDP
26.55
0.765
0.3074
103.17
25.80
0.681
0.3808
128.53
DDNM
28.80
0.822
0.2435
80.38
26.84
0.743
0.3083
102.82
DDRM
28.47
0.814
0.2511
84.87
26.43
0.730
0.3175
99.85
DiffRGD
27.49
0.778
0.2266
64.60
23.96
0.625
0.3687
121.27
Table 1: Main results on FFHQ and ImageNet using 100 images per dataset. Higher PSNR/SSIM and lower LPIPS/FID are better. Bold denotes the best result within each task and type.
Figure 2: Qualitative comparison of DPS-based methods on FFHQ across six representative inverse problems. Each panel shows two examples with the same original image and measurement. DPS+CAT denotes DPS equipped with CAT, while DPS+Diffstate is the corresponding DiffState baseline.
Figure 3: Robustness to the guidance step size ω on FFHQ motion deblurring. Each point aggregates 100 images.
Figure 4: Qualitative robustness to the guidance step size on FFHQ motion deblurring. Rows correspond to PSLD, PSLD+DiffState, and PSLD+CAT; columns show the ground-truth label, the blurred measurement, and reconstructions obtained with five values of ω . All entries use the same input image.
Figure 5: Robustness of PSLD+CAT to the tubular-radius coefficient ρ on FFHQ motion deblurring. Each point aggregates 100 images, and the horizontal dashed line marks the PSLD result reported in the main experiment.
Figure 6: Qualitative robustness of PSLD+CAT to the tubular-radius coefficient on FFHQ motion deblurring. Columns show the ground-truth label, the blurred measurement, the PSLD reconstruction from the main experiment, and PSLD+CAT reconstructions obtained with five values of ρ for the same input.
Figure 7: Robustness to the measurement-noise standard deviation σy on FFHQ motion deblurring. Each point aggregates 100 images.
Figure 8: Qualitative robustness to measurement noise on FFHQ motion deblurring. The left image shows the ground-truth label. Rows correspond to the four noise levels, while columns show the measurement and reconstructions from each method for the same input image.
Variant
N/T
Curv.
Tube
Armijo
DPS: Nonlinear deblurring
PSLD: Motion deblurring
PSNR ↑
SSIM ↑
LPIPS ↓
FID ↓
PSNR ↑
SSIM ↑
LPIPS ↓
FID ↓
Host
No
No
–
No
17.06
0.326
0.5698
214.46
17.96
0.366
0.6719
374.57
Host + Armijo
No
No
–
Yes
24.08
0.696
0.2751
71.71
25.74
0.695
0.3482
123.15
First-order CAT
Yes
No
1st
Yes
24.26
0.700
0.2718
73.44
25.80
0.699
0.3471
121.86
Tangent-only
Yes
No
Projection
Yes
24.15
0.696
0.2731
75.45
25.61
0.688
0.3504
125.81
CAT w/o Armijo
Yes
Yes
2nd
No
17.33
0.409
0.4555
192.73
19.18
0.413
0.6243
323.10
Table 2: Ablation of CAT on FFHQ using DPS for nonlinear deblurring and PSLD for motion deblurring. Each host and variant are evaluated on 100 images. “1st” and “2nd” denote first- and second-order tube models, respectively. Best results are shown in bold.
Figure 9: Qualitative component ablation on FFHQ. The first row uses DPS for nonlinear deblurring, and the second uses PSLD for motion deblurring. Each row shares the same ground truth and measurement across variants.
Study
Setting
PSNR ↑
SSIM ↑
LPIPS ↓
FID ↓
Memory (MiB)
Time (s)
Sampling steps
DPS
21.97
0.6269
0.3219
85.68
3490
38
N=200
23.50
0.6851
0.2922
82.91
3610
26
N=400
24.62
0.7152
0.2633
70.54
3610
51
N=600
25.42
0.7380
0.2432
64.24
3610
78
N=800
25.92
0.7525
0.2297
58.73
3612
103
N=1000
26.30
0.7623
0.2197
55.17
3612
132
Table 3: Efficiency of DPS+CAT on FFHQ nonlinear deblurring over 100 images using a single NVIDIA GeForce RTX 4090 GPU. Both blocks include the same DPS baseline for reference. The upper block varies the number of reverse sampling steps N with Armijo applied at every step, while the lower block fixes N=1000 and varies the Armijo period p . Memory and time denote the recorded GPU memory usage and wall-clock runtime, respectively. Bold marks the best reconstruction quality within each block.
Method
Observation time ratio: 100%
Observation time ratio: 10%
PSNR ↑
Blur PSNR ↑
χcp2↓
χlogca2↓
PSNR ↑
Blur PSNR ↑
χcp2↓
χlogca2↓
SMILI
22.67 (3.13)
27.79 (4.02)
1.878 (0.952)
17.612 (10.299)
20.85 (2.90)
25.24 (3.86)
1.209 (0.169)
21.788 (12.491)
EHT-Imaging
24.28 (3.63)
28.57 (4.52)
1.251 (0.250)
1.259 (0.316)
22.67 (3.46)
26.66 (3.93)
1.166 (0.156)
1.240 (0.205)
DPS
25.86 (3.90)
32.94 (6.19)
8.759 (37.784)
5.456 (24.185)
24.36 (3.72)
30.79 (5.75)
13.052 (43.087)
6.614 (26.789)
LGD
21.22 (3.64)
26.06 (4.98)
13.239 (17.231)
13.233 (39.107)
22.08 (3.75)
27.48 (5.09)
10.775 (21.684)
13.375 (56.397)
RED-Diff
23.77 (4.13)
29.13 (6.22)
1.853 (0.938)
2.050 (2.361)
22.53 (3.02)
27.67 (4.53)
2.488 (2.925)
4.916 (13.221)
Table 4: Black hole imaging on InverseBench at 100% and 10% observation time. Entries are mean (standard deviation). Baseline results are taken from InverseBench [ 24 ] ; DPS+CAT is evaluated on 100 test images. Bold marks the highest mean PSNR/Blur PSNR and the lowest mean reduced chi-squared within each observation setting.
Figure 10: Qualitative black hole imaging results for the same ground-truth source from InverseBench at 100% (top) and 10% (bottom) observation time.
Scale w
Method
CLIP ↑
FID ↓
Sat.
Contrast
10
CFG
31.6905
66.63
0.2901
0.2352
APG
31.4236
65.26
0.2014
0.1689
CFG+CAT (ours)
31.2664
64.67
0.2228
0.1943
20
CFG
31.8678
69.85
0.3538
0.2728
APG
31.7346
67.04
0.2212
0.1865
CFG+CAT (ours)
31.2742
65.59
0.2263
0.1974
Table 5: Classifier-free text-to-image guidance on 1000 COCO 2017 validation captions. Bold marks the best CLIP and FID at each guidance scale.
Figure 11: Qualitative comparison at guidance scale w=100 . Within each three-image group, all methods use the same prompt and initial seed. From left to right, the groups correspond to “A bird that is on a tree limb” and “A polar bear dives off a rock into a river.”
Appendix figures & tables3 assets
Supplementary material from the paper’s appendix.
Appendix
Dataset
Inverse problem
Host step
ρ
p
N
FFHQ
Random inpainting
4.0
0.1
1
1000
Box inpainting
2.0
0.1
1
1000
Super-resolution
2.0
0.1
1
1000
Motion deblurring
5.0
0.1
1
1000
Gaussian deblurring
5.0
0.1
1
1000
HDR reconstruction
10.0
0.1
1
1000
Appendix
Table 6: DPS+CAT and CFG+CAT hyperparameters used in the natural-image, scientific inverse-problem, and text-to-image experiments. The host guidance step corresponds to scale for natural images, step_size for InverseBench, and the CFG scale w for COCO. The symbols ρ , p , and N denote the tubular-radius coefficient, Armijo period, and denoising-step count, respectively. All CFG+CAT runs use a score-norm threshold of 10−8 and 30 bisection iterations with tolerance 10−4 .
Dataset
Inverse problem
γ
ω
ρ
p
N
FFHQ
Random inpainting
0.01
2.0
1.0
1
1000
Box inpainting
0.01
0.5
1.0
1
1000
Super-resolution
0.01
5.0
1.0
1
1000
Motion deblurring
0.01
5.0
1.0
1
1000
Gaussian deblurring
0.10
10.0
1.0
1
1000
ImageNet
Random inpainting
0.10
4.0
0.1
1
1000
Appendix
Table 7: PSLD+CAT hyperparameters used in the main experiments. The original PSLD hyperparameters are the latent data-consistency weight γ and guidance weight ω ; the remaining notation follows Table 6 . PSLD is evaluated on the five linear inverse problems supported by its host implementation.
Dataset
Inverse problem
sDPS
Ptt
ρ
p
N
FFHQ
Random inpainting
15.0
10
0.1
5
500
Box inpainting
5.0
500
0.1
5
500
Super-resolution
150.0
500
0.1
5
500
Motion deblurring
5.0
10
0.1
5
500
Gaussian deblurring
10.0
10
0.1
5
500
HDR reconstruction
30.0
500
0.1
1
500
Appendix
Table 8: Resample+CAT hyperparameters used in the main experiments. sDPS is the scale of the soft latent DPS step, Ptt is the time-travel period, and the remaining notation follows Table 6 .
Diffusion posterior sampling conditions diffusion priors on measurements, but data-consistency updates are typically scaled by hand-tuned guidance weights and can destabilize sampling under stiff, operator-dependent curvature. We replace scalar guidance with a per-noise-level damped Gauss--Newton correction computed in diffusion-state coordinates. The correction pulls likelihood gradients back through the denoiser, uses a one-sided curvature model that avoids forward denoiser Jacobians, and applies diffusion-calibrated rank-one damping aligned with the denoiser residual. Each correction is solved with matrix-free GMRES using automatic differentiation, and sampling proceeds with a variance-preserving Langevin transition with a closed-form drift/noise split. On FFHQ and ImageNet across inverse problems, it achieves competitive PSNR/SSIM/LPIPS while running markedly faster than most of the compared baselines; on accelerated MRI reconstruction, it achieves the best PSNR/SSIM among the compared baselines.
Seunghyeok Shin, Minwoo Kim, Dabin Kim +1
Department of Electrical and Computer Engineering, Inha University, Incheon, 22212, South Korea.
Diffusion models generate samples by learning to reverse a fixed corruption process, and classifier-free guidance (CFG) is the standard mechanism for conditioning this process on a desired class or prompt. CFG can be applied at varying guidance strengths, and while higher strengths improve image quality and conditional alignment, too high a guidance strength can degrade image quality and diversity. Furthermore, CFG violates principled diffusion sampling dynamics, and existing explanations for why it works despite the violation disagree on the underlying theory or do not extend to deterministic samplers used in practice. We address both these issues. We first frame CFG sampling as a continuous-time optimal control problem, treating the sampling trajectory as a sequence of controls chosen to maximise the probability of the desired condition. Solving the resulting Hamilton--Jacobi--Bellman equation shows that CFG is recovered under specific path costs when using an unconstrained control set. We argue this lack of constraint is responsible for CFG's failure at high guidance strengths, since it permits the sampling path to move arbitrarily far from the current image estimate. To fix this, we propose curvature-aware CFG (CACFG), which constrains the control set to a hypersphere informed by the Gaussian regularisation used when training variational autoencoders. We show that the control inputs produced by CFG sampling routinely violate this bound, and that across diffusion models, datasets, and guidance schedules, CACFG achieves superior generative quality at mid-to-high guidance strengths with a less severe quality-diversity tradeoff than regular CFG.
Max Collins, Dasith de Silva Edirimuni, Jordan Vice +2
School of Physics, Maths and Computing The University of Western Australia Perth, WA 6009
Classifier-Free Guidance (CFG) has established the foundation for guidance mechanisms in diffusion models, showing that well-designed guidance proxies significantly improve conditional generation and sample quality. Autoguidance (AG) has extended this idea, but it relies on an auxiliary network and leaves solver-induced errors unaddressed. In stiff regions, the ODE trajectory changes sharply, where local truncation error (LTE) becomes a critical factor that deteriorates sample quality. Our key observation is that these errors align with the dominant eigenvector, motivating us to leverage the solver-induced error as a guidance signal. We propose Embedded Runge-Kutta Guidance (ERK-Guid), which exploits detected stiffness to reduce LTE and stabilize sampling. We theoretically and empirically analyze stiffness and eigenvector estimators with solver errors to motivate the design of ERK-Guid. Our experiments on both synthetic datasets and the popular benchmark dataset, ImageNet, demonstrate that ERK-Guid consistently outperforms state-of-the-art methods. Code is available at https://github.com/mlvlab/ERK-Guid.
Inho Kong, Sojin Lee, Youngjoon Hong +1
Korea University · KAIST · Seoul National University +1