Kernel Density Steering: Inference-Time Scaling via Mode Seeking for Image Restoration
Authors: Yuyang Hu
Kangfu Mei
Mojtaba Sahraee-Ardakan
Ulugbek S. Kamilov
Peyman Milanfar
Mauricio Delbracio
Venue: NeurIPS 2025 poster ()
Keywords: Diffusion Model
Image Restoration
Image Processing
Primary area: deep_learning

Abstract: Diffusion models show promise for image restoration, but existing methods often struggle with inconsistent fidelity and undesirable artifacts. To address this, we introduce Kernel Density Steering (KDS), a novel inference-time framework promoting robust, high-fidelity outputs through explicit local mode-seeking. KDS employs an $N$-particle ensemble of diffusion samples, computing patch-wise kernel density estimation gradients from their collective outputs. These gradients steer patches in each particle towards shared, higher-density regions identified within the ensemble. This collective local mode-seeking mechanism, acting as "collective wisdom", steers samples away from spurious modes prone to artifacts, arising from independent sampling or model imperfections, and towards more robust, high-fidelity structures. This allows us to obtain better quality samples at the expense of higher compute by simultaneously sampling multiple particles. As a plug-and-play framework, KDS requires no retraining or external verifiers, seamlessly integrating with various diffusion samplers. Extensive numerical validations demonstrate KDS substantially improves both quantitative and qualitative performance on challenging real-world super-resolution and image inpainting tasks.

Kernel Density Steering: Inference-Time Scaling via Mode Seeking for Image Restoration
Yuyang Hul.2- Kangfu Mei', Mojtaba Sahraee-Ardakan', Ulugbek S. Kamilov² Peyman Milanfar', Mauricio Delbracio
'Google, Washington University in St. Louis (kangfumei, jabaaa moojjaabammilanaaa milamfar ar mdelbra/ Ggoogle com yuyang, kami lov kanstor
Abstract
Diffusion models show promise for image restoration, but existing methods often struggle with inconsistent fidelity and undesirable artifacts. To address this, we introduce Kernel Density Steering (KDS), a novel inference-time framework pro- moting robust, high-fidelity outputs through explicit local mode-seeking. KDS employs an N-particle ensemble of diffusion samples, computing patch-wise ker- nel density estimation gradients from their collective outputs. These gradients steer patches in each particle towards shared, higher-density regions identified within the ensemble. This collective local mode-seeking mechanism, acting as "collective wisdom", steers samples away from spurious modes prone to artifacts, arising from independent sampling model imperfections, and towards more robust, high- fidelity structures. This allows us to obtain better quality samples the expense higher compute by simultancously sampling multiple particles. As plug-and-pn framework, KDS requires no retraining or external verifiers, scamlessly integrating with various diffusion samplers. Extensive numerical validations demonstrate KDS substantially improves both quantitative and qualitative performance on challenging real-world resolttino and image inpainting tasks.
Introduction
Image restoration (IR) seeks recover high-quality image from its degraded observation y. Recently, diffusion models (DMs) [1-3] have been successfully applied to challenging IR tasks [4-8], including super-resolution (SR) [9-14], image deblurring [15-17], and image inpainting [18-20]. DMs operate on a dual process: a forward process introduces noise to a clean image x over timesteps, creating sequence of noisy latents where zr is approximately pure noise. A learned reverse process then iteratively refines these latents from zy back to an estimate of zg-
To enable DMs for IR task, straightforward way is train diffusion models directly conditioned on information derived from the low-quality image [21-23]. Conditioning can be implemented through various techniques, for example, input concatenation [21], cross-attention [24, 25]. adapters like ControlNet [26] or LoRA [27]. This allows the model's 's network c0(2,, f, c) to estimate the con- ditional score function V loge: (2//c) to directly approximate target score function V logpe yy) enabling standard sampling at inference.
A key challenge here lies in navigating the perception-diitortion trade-off [28]. Standard posterior sampling can yield perceptually sharp results, but often suffers from high distortion and hallucinations. Conversely, averaging multiple independent samples approximates the Minimum Mean Squared Error (MMSE) estimate, minimizing average distortion but leading to blurriness and poor perceptual
"This work was done during an internship at Google.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Standard Dillsion Bamplnn
Kamal Danutty Eshanced
Independent Sumpling
Kemel Densiy Steering
Time
Donsity
Time
Denaity
Figure 1: Conceptual illustration of Kernel Density Steering (KDS) for diffusion sampling. Left: Standard diffusion sampling with N independent particles (blue dots) can result in high variance. Right: KDS utilizes the ensemble of N particles to estimate local density. It then guides these particles towards shared high-density modes (peaks in the density curve) during the diffusion process.
quality [29]. Neither extreme ideal for high-quality IR. Many existing methods improve this trade-off by additional model training or specialized network architectures [4, 29-34j.
This paper introduces Kernel Density Steering (KDS), inference-time framework designed to improve the cetortiionpprcepttoon trade-off without requiring any model retraining. KDS leverages collective information from an N --particle ensemble by steering particles towards regions of high concentration within the predicted solution ensemble.
This mechanism is implemented as two-step process. First, at each step i, the denoiser predicts the clean data 2 for each particle i. Second, it performs non- non-parametric mode-seeking operation to steer each prediction 2 toward local region of high density. This is achieved by implicitly estimating the ensemble's 's density using Kernel Density Estimation (KDE) [35, 36] and then applying a single mean-shift update-which which is equivalent gradient ascent step this estimated density- to
find a consensus point 20 (().
Σ
-==1"
N
K
|
2
(k)
(1)
(1) 2n 112
(10
k= K
This refined estimate z defined as:
defines a new consensus score, $ The original and consensus scores are
Vanzo and Z (i)
of
of
This guidance is incorporated into the probability flow ODE via weighted average of the two scores:
dz (
1 [()))) - go(
(1=))ss()) + 5,5(1) dt,
where 6, is the KDS guidance strength. Thus, particles are guided by the original score and the KDS term. This term functions as a form of collaborative filtering, using non parammeetrii mode-seeking to guide the ensemble toward areas of high consensus. This adaptive, internal guidance aims for restorations that are both quantitatively accurate and perceptually sharp.
To demonstrate the effectiveness of this approach, we first evaluate KDS's impact on posterior sampling in a controlled synthetic task. Subsequently, we provide extensive empirical validation, showing significant improvements in both quantitative performance and qualitative robustness challenging real-world SR and inpainting tasks.
Our contributions are: (1) We introduce Keree Density Steering (KDS), a novel ensemble-basee inference time guidance method that uses patch-wis wish KDE gradients to steer diffusion sampling distortion-perceptii perception trade-off in image towards high-density posterior modes. (2) KDS improves the distortion
restoration by operating directly on the sampling process without retraining. (3) We design a patch- wise mechanism to overcome the curse of dimensionality, enabling KDS to scale to high-dimensional latent space. (4) Empirical validation showing significant improvements in fidelity, perception, and robustness challenging real-world super-resolution and inpainting tasks.

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2 Background
2.1 Latent Diffusion Models
Latent Diffusion Models (LDMs) [24] efficiently perform image generation by operating in a lower- dimensional latent space learned by a pre-trained autoencoder [37], mitigating the computational cost of pixel-space methods on high-resolution images. The autoencoder includes an encoder mapping an image æ to a latent
{ and a decoder 'D mapping 2 back to Ø ((z).
The forward diffusion process gradually adds Gaussian noise to the target latent sample 20 = (x0) over 7 timesteps according noise schedule Oze, This process results in the following marginal distribution for the latent 2, at any timestep I:
(<z1|20) == (zz; Vārzu, (1 (1--0j) T.) where de II 0s5
(4)
The reverse process aims to generate clean latent variable 2o from noisy latent variable øy (0, I). This is achieved by training a neural network cø(2,.
(z, 1) to predict the noise component 6 that was added during the forward process. This predicted noise is directly related to the score function of the noisy latent distribution pt (z)): Vz, logp: R VI-a co co(2: [zt, t). By accurately predicting the noise, the network learns to estimate this score function, thereby implicitly capturing the data distribution p(ze) at different noise levels.
To sample from the learned distribution, Denoising Diffusion Implicit Models (DDIM) [38] offer fast, non-Markovian sampling procedure. The sampling step from 2t to z DDIM, which predicts the latent state at - -1 based on the estimated clean latent at time t, is given by:
Zt VI Cr.co t,
2t-1 v
+ / 1c
(5)
Vā
At the end of sampling, the predicted za is then decoded by D to obtain the generated image do.
2.2 LDM for Image restoration
Two primary methods exist for leveraging diffusion models in IR. Conditional LDMs [4-6, 9- 11, 15, 16. 18-20, 391, learn the posterior p(=lc) directly, where the conditioning e is derived from y. These models can be highly effective but may face challenges in accurately modeling the true posterior for complex degradations, potentially impacting restoration fidelity. Alternatively, unconditional LDMs can be employed with inference-ting guidance using Bayes' rule [40-45 45-45). These methods typically require computing gradients of the log- ligellihoooo logp(y) y ax,) (often approximated via the estimate kojz). This necessitates knowledge of the degradation process (e.g., the operator A in y A Arr), limiting their applicability in scenarios where the degradation is unknown or cannot be accurately modeled which is the case in many real-world image SR or deblurring.
2.3 Inference-Tinee Scaling in Diffusion Models
Merely increasing the number of denoising steps (NFEs) for single sample yields diminishing returns [46], motivating research into more effective inference- time strategies [47, 48). One prominent avenue, particularly explored in Text-to- Image (T2I) generation, involves actively searching for optimal initial noise vectors (2T) or resampling generation trajectories, often guided by external verifiers (e.g., CLIP score, pre-trained classifiers) build a reward function and select promising candidates [47, 48). However, this introduces reliance on these external verifiers, which can suffer from biases misalignment with the true image restoration (IR) target.
second class of strategies are specifically designed for imaging inverse problems. These often employ ensemble methods, such those based on Sequential Monte Carlo (SMC) techniques [49- 52], employ likelihood approximation and reweighting sampling trajectories to achieve more accurate approximation of the posterior distribution p( (y) with unconditional DMs. These approaches, however, require accurate knowledge of the degradation model specific to the IR task. This assumption often invalid in many real-world applications, such as real-world image SR and deblurring.
In contrast to both these approaches, KDS introduces an ensemble method that operates without externl verifiers or the need for the knowledge of degradation model. By focusing internal
3

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consensus guidance derived directly from particle interactions within the ensemble, KDS offers more robust and broadly applicable solution for enhancing image restoration quality.
2.4 Mean Shift for Mode Seeking
Mean Shift [53, 54] is a non-parametric clustering and mode-seeking algorithm. Given a set of points (æx ) the goal is to find the modes (local maxima) of the underlying density function. The algorithm iteratively shifts each point towards the local mean of points within its neighborhood. weighted by kernel function (often Gaussian). For point a and kernel K with bandwidth h, the mean shift vector is calculated as:
EN
K
2 12
m(x)
(6)
K a
12
km
crucial property of the mean shift vector, m(x). is its direct proportionality to the gradient of the logarithm of the Kernel Density Estimate (KDE): m(x) x Vologåk (z). Therefore, iteratively updating a point by adding its mean shift vector (æ + æ + m ()) an effective method for performing gradient ascent the KDE, guiding the point towards a local mode of the estimated density. This established connection is leveraged in our proposed KDS framework.
Kernel Density Steering for Diffusion Samplers
Kernel Density Steering (KDS) is inference-time technique for Image Restoration that enhances diffusion model sampling by leveraging an N ---patticle latent ensemble, Z: = {z, } Standard diffusion sampling result in outputs with inconsistent fidelity and undesirable artifacts. KDS addresses this by guiding all particles towards regions of high consensus within their own ensemble which is hypothesized to lead more robust and high-fides fidelity restorations.
KDS achieves this ensemble-drives driven mode-seeking by incorporating an additional steering term into the sampling update for each particle. This steering is inspired by the Mean Shift algorithm (Sec. 2.4).
Specifically, for each particle, KDS computes mean shift vector m(z (6)) based on the current ensemble Z, Then KDS steering is applied by taking a step in the direction of this mean shift vector, which directs local mode of the ensemble's Kernel Density Estimate (KDE).
3.1 Patch-wise Mechanism and Integration
Directly applying KDE and mean shift in the full, high-dimensional latent space 2, is not achievable. prohibitively large number of particles would be needed to obtain reasonable density estimate via KDE in such high-dimensional spaces. To address this, we introduce patch-wise mechanism to apply the KDS principle smaller, more manageable, lower-dimensional patches. These patches are extracted from the predicted clean latent state =(()) corresponding to each particle 2{{) in the
(l)
ensemble. This predicted clean latent, is the output of the diffusion model's denoising step at time t, obtained via:
(4) 1 Off
i VI- azco(z) <ioo(z ,t, ct).
V az
Patch-wise Mechanism: To the ensemble of predicted clean latent states 2 of our patch-wise mechanism involves the following steps at each timestep l:
1. Patch Extraction: For each predicted clean latent S// in the ensemble (obtained from Eq. 7),
extract a set of non-overlapped patches (p, (1)) where we have omitted the time index to simplify the notation. This yields an ensemble of corresponding patches P, = { spatial patch location j.
(k) 1N for each given
2. Compute and Apply Patch-wise Steering: For each patch p) (4) (from particle at location j), its mean shift vector m(p,)) (6) is first computed based on the local ensemble of corresponding patches

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P-= )e This vector, which directs the patch towards a region of higher consensus within the ensemble, is given by:
Σ
G
[Ip
(9)
-p,
12
(2)
Pj
(k)
m(p)")
P3
Σn G
1pp P.
Here, G is a Gaussian kernel function and h is the bandwidth hyperparameter. As discussed in Sec. 2.4, this vector m(p)") points from the current patch towards the estimated local mode of its ensemble's density. This computed mean shift vector is then used update the patch:
(i) t " + 6,m(p))) )
Pj
where de E [0, 1] is a time-dependent steering strength.
Reconstruct Guided Latent Prediction: After all patches {p/ from given #0): have been updated to (p)).kDs) ), they are reassembled by direct replacement to original coordinate to form the KDS-refined clean latent prediction, (D). Df/
Integration with Diffusion Samplers: KDS integrates seamlessly with standard diffusion samplers (e.g., first-order ODE DDIM solver and high-order ODE DPM-Solver++ [55]). At each sampling step 1, the sampler first computes the predicted clean latent for each particle, a(f) using (Eq. 7).
Patch-wise KDS is then applied to this ensemble of predictions (+) 1 to produce the ensemble of
refined predictions {z
(1).DDS ) The sampler subsequently uses these KDS-refined estimates to
Ji=1-
compute the next latent states {- {2t-1hi11 After the full reverse diffusion process, an ensemble of 1
(DS I is obtained.
KDS-guided clean latent states {20
Final Particle Selection: To produce a single, representative output image from an ensemble of (6).KDS J
generated candidates [#)
ii=1 selection is made in the latent space. Our proposed strategy is
to choose the latent vector from the ensemble that closest to the ensemble's mean Eg:
exlleced arg min [| žo113-
(10)
This selected latent vector 20 2gclected is then transformed by the decoder into the final image Sp. as represented by the equation Xg D(2gglece))
4 Experiments
In this section, we present experiments to numerically evaluate our proposed method. First, we utilize 2D toy example (Sec, 4. to visually illustrate the impact of our proposed Kernel Density Steering on the diffusion model sampling process, providing intuition for its mechanism. Subsequently, we demonstrate the effectiveness and practical applicability of our approach on two challenging real-world tasks: image super-resolution (SR) (Sec. 4.2.1) and image inpainting (Sec. 4.2.2).
4.1 KDS Sampling in a 2D Example Case
To visualize KDS's effect, we consider sampling from a 2D Mixture of Gaussians (MoG) target distri- bution p(z.) >en (xNV M4: Σ.). In this controlled setting, the exact score V (x;) of the noisy distribution is known. KDS is applied by adding its steering term (derived from Eq. 9 using (i)
the 2D particle states (x(") J.N ,) to the update driven by the exact score within DDIM sampler. As the data dimension here is very low, we let the patch size equal to the full data size.
As illustrated in Figure 2, KDS significantly sharpens the resulting sample distribution. Samples cluster more tightly around the true mode means (Ju.), effectively reducing the spread (variance) of samples within each mode compared to using the exact score alone. This occurs because KDS actively guides particles towards their respective mode wenicrs-which correspond to regions of high sample density in the ensemble-then thereby improving concentration of samples around each mode.
5

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0T-0.0
DDIM
DDIM KDS
Figure 2: Kernel Density Steering (KDS) sharpens sample distributions in 2D Mixture of Gaussians (MoG) toy problem. The target distribution p(zo) consists of three distinct Gaussian modes. Samples drawn using DDIM with the exact score function (purple dots) versus DDIM augmented with KDS (green dots, N = 50 particles). KDS guides particles through the reverse diffusion progresses, leading to significantly higher sample concentration the mode peaks compared standard DDIM.
Table 1: Super-Resolution Performance on DIV2K dataset with DDIM (50 steps)
Method
PSNR
SSIM
LPIPS (L)
NIMA
DISTS (1)
MANIQA
CLIPIQA
FID (4)
LDM-SR
22.05
0.531
0.307
4.922
0.179
0.390
0.594
20.89
+ KDS (NV 10)
22.37*
0,549*
0.292*
4.949*
0.176*
0.399x
0.601
20.78*
DitBIR
21.56
0.488
0.377
5.195
0.223
0.566
0.730
32.28
+ KDS (NV 10)
22.44x
0,535*
0.348*
5.219*
0.220*
0.571
0.744*
30.67*
SceSR
22.43
0.573
0.340
4.902
0,200
0.423
0.605
25.96
+ KDS (N= 10)
22.79x
0,587
0,313
5.026*
0.191
0.488*
0.679*
25.44x
Statistically significant differences (P 0.05) compared with DDIM marked with
4.2 Real-world Image Restoration
To demonstrate the effectiveness of our Kernel Density Steering (KDS), we compare its performance against baseline sampling method image super-resolution and inpainting tasks. Our experiments utilize two widely used samplers (DDIM, DPM-Solver++). For each configuration, we contrast the results obtained with standard sampling versus KDS-enhanced sampling, ensuring fairness by using the same initial noise. We compare performance with and without KDS, using the same initial N noise (2) Ti=! for fairness across ensemble of N 10 particles unless specified otherwise. Full experimental details, including hyperparameter settings (e.g., patch size, steering strength 5,) and comprehensive ablation studies, provided in the Appendix.
Table Super-Resolution Performance on real-world collected datasets with DDIM (50 steps)
Datascts
RealSR
DrealSR
Method
PSNR
SSIM
LPIPS (J)
DISTS ()) CLIPIQA
PSNR
SSIM
LPIPS (J)
DISTS (L) CLIPIQA
LDM-SR
24.01
0.666
0.308
0.211
0.624
26.13
0.689
0,342
0.222
0.610
KDS (N = 10)
24.68+
0.703+
0.275+
0.201
0.622
26.32*
0.702+
0.331+
0,317+
0.617+
DimBBR
23,21
0.610
0.370
0.250
0.689
23.99
0.551
0.491
0.293
0.701
KDS (N = 10)
24.39+
0.669+
0.339+
0.244+
0.692+
25.63
0.645
0.422+
0.276+
0.693+
SecSR
24.29
0.710
0.279
0.204
0.588
26.97
0.750
0.300
0.218
0.625
KDS (N = 10)
24.50*
0.719x
0.272*
0,206*
0.640x
27.42x
0.765*
0.287+
0.212*
0,651x
Statistically significant differences (P 0.05) compared with DDIM are marked with a
4.2.1 Real-world Image Super-Resolution
We evaluated the performance KDS for 4x real-world super-resolution (SR). This evaluation uti- lized the LDM-SR [24], DiffBIR [4], and SeeSR [33] backbones across several datasets: DIV2K [56], RealSR [57]. DrealSR [58]. Performance was evaluated using a comprehensive suite of metrics,
including PSNR, SSIM [59], LPIPS [60], FID [61]. NIMA [62], MANIQA [63], and CLIPIQA [64].
Results: Quantitative results consistently show KDS's benefits. Tables and Tables 2 demonstrate that adding KDS significantly improves both distortion and perceptual metries across different

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LDM-SR
LR
w/o KDS
w/ KOS
Reference
LR
w/o KDS
w/ KDS
Reference
DiffBIR
LR
w/o KDS
w/ KDS
Reference
Reference
Faa G
(p)))))))))
(i)
eduroam oduroam eduroam eduroam
SeeSR
LR
w/o KDS
w/ KDS
Reference
LR
w/o KDS
w/ KDS
Reference
RICS PMN
R0CS PM
HIRD'S
JBC' PMN
HDID'S
JOSS PM
K'S
K'S
K'S
K'S
Figure 3: Qualitative comparison of 4x Real-world Super-Resolute with KDS-enhanced DDIM sampling (Number of Particles N = 10). "w/o KDS' shows results from baseline DDIM, while 'w/ KDS' shows results with KDS. KDS consistently produces images with improved sharpness, finer details, and reduced artifacts across LDM-SR, DiffBIR, and SeeSR backbones.
datasets, backbones, and samplers (DDIM, DPM-Solver+++) compared the respective baselines without KDS. Qualitative results in Figure 3 corroborate these findings, showing restorations with higher fidelity and fewer artifacts when using KDS.
Compare with Best-of-N approaches: While inference-time scaling allows for the generation of multiple candidate solutions, especially beneficial for low-quality inputs, selecting the optimal one from an N-particle ensemble is non-trivial. As Table shows, using a non-reference metric like LIQE [65] to pick the best particle results significantly lower performance compared our kernel density steering (KDS) method. This demonstrates that, despite comparable computational costs, traditional post-sampling selection methods do not achieve the same level of performance as KDS.
Table 4: Comparison to Best-of-N (BoN) using LIQE [65] on RealSR with DiftBIR.
Table 3: Inpainting Performance on ImageNet
with LDM-inpainting backbone.
Method N PSNR LPIPS (4) Time Memory
Method
PSNR SSIM LPIPS(J) FID())
DDIM
23.21
0.370
7.9s
8.0Gb
DPM-Solver
19.94
0.725
0.144
11.70
KDS (N=10) 10) 21.29 0.747
0.135
11.29
BoN 5 23.72
0.361
14.9s
9.7Gh
+ KDS
24.17
0.349
15.6s
9.7Gb
DDIM
21.03
0.736
0.140
11.47
KDS (N = 10) 21.35
0.748
0.131
11.18
BoN 10 23.77
0.366
27.3s
10.4Gh
+ KDS 10 24.39
0.339
28.4s
10.4Gb
4.2.2
Image Inpainting
We evaluate the performance of KDS (N=10) on a center box inpainting task using the ImageNet dataset. For this task, the central 30% square region of each image masked. employ Latent Diffusion Model (LDM) for inpainting, initialized with weights from pretrained text-to-image model that matches the architecture and size of Stable Diffusion v2 [24]. The LDM was fine-tuned on ImageNet dataset for random box inpainting. The training utilized batch size of 1024 and

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Comparison
Comparison
06.1
303
27.3
25.
parts
Figure 4: Robustness analysis of KDS on the RealSR dataset. Left: Scatter plots comparing the PSNR and LPIPS the DDIM with KDS versus worst- porso-peroonnnng particle of standard DDIM ensemble (N = 10). KDS consistently improves the quality of the worst-case samples. Right: Qualitative examples comparing the porst-perfomming output from DDIM ensemble with KDS-guided DDIM with the same random seed, demonstrating KDS's s superior consistency and artifact reduction.
LDM-Inpainting
Input
Input
DPM BPM-Solver
DPM Solver KDS
Reference
Input
DPM-Salver
RPM-So KDS
Reference
Figure 5: Box-inpainting performance of KDS. KDS generates more coherent and detailed inpainted regions compared standard DDIM sampling.
learning rate of le-4, parameters empirically selected maximize compute efficiency, Performance was measured using PSNR, SSIM, LPIPS, and FID.
As shown in Table 3, our proposed KDS improves quantitative inpainting performance. Furthermore, Figure 5 demonstrates that KDS generates inpainted regions with enhanced fidelity, greater coherence with surrounding image content, and more plausible details compared standard sampling.
4.3 Analysis of KDS Robustness
key objective of KDS is to enhance the reliability and consistency of diffusion based restorations, particularly by reducing artifacts and improving fidelity.
Figure 4 demonstrates this improved stability. The seatter plots (left) compare the performance (PSNR and LPIPS) of DDIM enhanced by KDS against the worst-performing particle from baseline DDIM ensemble (also N = 10) on RealSR dataset. Points above (PSNR) or below (LPIPS) the y line signify that KDS improves the worst-case performance. These two scatter plots clearly demonstrate enhanced restoration reliability, Qualitative examples (right part of Figure 4) further illustrate KDS's 's consistency and artifact reduction.
4.4 Impact of KDS Hyperparameters: Number of Particles and Bandwidth
The performance KDS, as an ensemble-based technique, inherently linked to its key hyper- parameters: the number of particles N and the kernel bandwidth h. We analyze their impact SR performance (PSNR, SSIM, LPIPS, FID) using the LDM-SR model on the DIV2K dataset, as illustrated in Figure 6.
Number of Partieles N: Increasing the ensemble size N consistently enhances restoration quality. As shown in Figure 6, both distortion metrics (e.g., PSNR, SSIM) and perceptual metrics (e.g.. I.PISS, FID) benefit, particularly as increases from small (e.g., N Rs 5- 10), where kernel density estimates are less robust, moderate (c.g., N x 15 20) values. While further increasing N (e.g..

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754A Number Particas (VPP)
Number Tartddes (M77)
JPPS us. humber Partoi (MP7)
va Partiddas (hF7)
23.5
4422
22.5
6:20
22.
Figure 6: Influence of Particle Number (N) and Bandwidth (h) Real-world SR. Performance metrics (PSNR, SSIM, LPIPS, FID) are plotted against the number of particles (N) for different bandwidth (/) settings on the DIV2K dataset, using LDM-SR backbone with KDS.
DDIM
DDIM KCS (A = 0.1)
DDIM KDS 0.3)
DDIM KDS (A =
DDIM Averaging
Figure 7: Influence of Bandwidth (h) on Real-world SR based on LDM-SR backbone. Note, DDIM + averaging means post-sampling averaging across the whole ensemble.
beyond N > 15 in depicted scenarios) can yield additional improvements, the gains may become marginal, leading to saturation. This underscores the inherent trade-off between the computational cost, which scales linearly with N, and the achievable performance boost.
Kernel Bandwidth h: The kernel bandwidth h plays crucial role in balancing the perception- distortion trade-off. Figure 6 (different colored lines representing different f settings) demonstrates that while larger ensembles generally improve performance, the choice of is critical. Excessively large bandwidths (e.g.. / 0.6 in some cases) can lead to over-smoothing, potentially enhancing distortion metrics like PSNR & SSIM the cost of perceptual quality (higher LPIPS FID). Con- versely, very small h might not capture sufficient local consensus. Optimal h selection is therefore essential for achieving the desired balance between quantitative accuracy and perceptual sharpness. Our experiments suggest that a moderate h often provides a good compromise. Furthermore, in Appendix A.2, we show that standard automatic bandwidth selection strategies (e.g., median heuristic) achieve performance nearly identical to our manually tuned h. This demonstrates that KDS can be deployed effectively without tedious manual tuning, enhancing its practical applicability.
Further parameter analyses in Appendix A.2 confirm KDS is robust to hyperparameter choices, including different kernel functions (e.g., Gaussian, Laplacian). We also explore the trade-off between particle count (N) and patch size. As shown in the appendix, increasing N enables the use of larger patch sizes, which enhances performance by better mitigating the curse of dimensionality.
Discussion
We further analyze KDS by comparing it to other inference-time techniques. We compare it against other mode-seeking sampling strategies (HDS [66], DGS [67]) and separately against other methods, like InDI [29], which are also designed to improve the perception-diitortion trade-off
Comparison with HDS [66] and DGS [67]. To position KDS relative other inference-time mode- secking techniques, we compare it against High Density Sampling (HDS) [66] and Density-Gune -uidded Sampling (DGS) [67]. We implemented both methods with the DR [4] backbone and swept their key hyperparameters (time for HDS, quantile for DGS) to find their best- configurations on the RealSR dataset, with full parameter sweep available in Appendix A.2.
As shown in Table 5, HDS achieves high PSNR severe cost to perceptual quality (LPIPS, CLIPIQA), while DGS offers moderate improvement, In contrast, KDS (N 10) provides the best balance, uniquely achieving both a strong PSNR and the best perceptual scores (LPIPS, DISTS, CLIPIQA), confirming its superior perception-distoriin trade-off.

---

The core conceptual difference lies in their assumptions, Both HDS [66] and DGS [67] are mode- seeking but rely on different assumptions compared with KDS. HDS assumes a unimodal Gaussian distribution, which is often violated by highly multi-modal IR posteriors. DGS, while non-parametric in different sense, guides single sample by progressively sharpening the marginal density pe (x)) at each step, forcing it toward a single high-density mode. In sharp contrast, KDS makes explicit assumption about the posterior's shape. uses a non- con-paamettri KDE on an ensemble of outputs (Zuje), making it better suited to finding a consensus within the multi-modal posteriors common in IR tasks.
Comparison with InDI [29]. We also distinguish KDS from InDI [29], which also adjusts the distortion perception trade-off at inference time. Their core approaches differ: InDI is not a posterior mode-seeking algorithm. It performs progressive refinement to the final estimation Ep. which does not aim to sample from the posterior mode. KDS, in contrast, directly seeks a consensus mode from the final clean output ensemble (εn). Moreover, KDS is training-free, plug-and-play module, whereas InDI requires an integrated training and sampling pipeline.
Table 5: Comparison with other Mode-Seeking Methods on RealSR dataset with DiffBIR backbone.
Method
PSNR Lo
SSIM Lo
LPIPS (4) Lo
DISTS (4) do
CLIPIQA Lo
DiABIR
23.21
0.61011.0015
0.370:10.011
0.250:10.005
0.689-10.013
+ HDS [66]
24.56=0..344
0.386=0.017 0.646=0.020+
0.274=:0.016
0.647=0.025+
DGS [67]
24.05
0.651
0.355
0.246
0.687+:0.013
+ KDS (ours)
2.239
0.339-+0.01 0.669
0.244410.004x 0.692+
Statistically significant differences (P < 0.05) compared with DDIM are marked with
Limitations & Future Work
Limitations. The primary limitation of KDS is its computational overhead, which scales linearly with
particle count N. Although 10 - 15 provides a good performance-coss trade-off, this overhead could be mitigated by future work on adaptive ensemble sizes or more efficient KDE techniques.
Future Work. As training-free ffree, plug-and-play framework, KDS is highly generalizable. Promising directions include applying it to other inverse problems (c.g., deblurring, denoising, medical imaging) and video restoration, where its patch-wise mechanism could be extended 3D spatio-temporal patches to enhance temporal consistency.
7 Conclusion
This work introduces Kernel Density Steering (KDS), a novel inference-time framework that enhances the fidelity and robustness of diffusion-based image restoration. The core of KDS lies in its local mode seeking strategy: it utilizes an N-particle ensemble and computes patch-wise kernel density estimation (KDE) gradients from predicted clean latent patches. These gradients guide samples towards shared, high-density regions, effectively steering them away from spurious modes to produce robust, high-fidelity restorations. As plug-and-plat and-play approach. KDS requires no retraining, external verifiers, or explicit degradation models, making it broadly applicable.
Acknowledgements
The authors would like to thank our colleagues Keren Ye, Viraj Shah, Ashwini Pokle and Mo Zhou for reviewing the manuscript and providing valuable feedback.
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A Supplementary Material
This supplementary material provides further details to the main paper. It covers the following aspects:
Section A.1: Experiment Details for Real-world Image Super-Resolution: Including dataset descriptions, backbone model configurations, an in-depth discussion of the patch- wise KDS mechanism with ablation studies, interaction of KDS with various diffusion samplers (DDIM, DPM-Solver+++) and additional comparative results severe degradation datasets.
Section A.3: Experiment Details for Image Inpainting: Covering the datasets used and the hyperparameter settings for the inpainting task.
Section A.4: Additional Visual Results: Presenting more qualitative examples for both super-resolution and image inpainting tasks to further demonstrate the efficacy of KDS.
A.1
Experiment details: Real-world Image Super Resolution
A.1.1
Datasets
Our real-world image super-resolution experiments utilized synthesized dataset derived from
DIV2K [56], alongside two real-world datasets: RealSR [57] and DRealSR [58]. For the DIV2K-
based synthesized data, we used the same dataset provided by StableSR [11], which consists of
3,000 randomly cropped patches (resolution: 512 x 512 each) from the DIV2K validation set [56].
Subsequently, we generated corresponding row-resolution (LR) images (resolution: 128 128)) using the degradation model adopted in Real-ESRGAN [68]. For the RealSR [57] and DReatSR [58] datasets, we adhered to the protocol from [11] to center-crop the provided LR images to 128 x 128.
A.1.2 Backbone Model Configurations
To assess the effectiveness of our proposed KDS method, applied it three established backbone models: LDM-SR [24], DiffBIR [4], and SeeSR [33]. For LDM-SR, the LR image was directly employed conditional input, concatenated with the LDM's primary input. For DifIBIR [4] and SeeSR [33], we adopted the hyperparameter settings recommended on their official GitHub pages. This included their specified parameters for text prompting, classifier-free guidance, and the use of their official pre-trained model checkpoints.
A.1.3 Patch-wise KDS Mechanism: Configuration and Discussion
As introduced Section 3. we employ a patch-wise mechanism to facilitate mode-seeking within the high-dimensional latent space. The motivation is that the accuracy of the mode-secking process is intrinsically linked to both the number of samples available for kernel density estimation (KDE) and the dimensionality of these samples. In case, direct mode-seeking vector z (which encodes features for the 512 x 512 x 3 image space) is impractical with a practical number of particles (e.g., 10-20).
the entire 64 x 64 x 8 latent
Patch Size Configuration: To effectively implement our patch-wis wise strategy, the patch size is chosen to balance accurate mode-seeking with computational efficiency. Accurate mode-seeking across the entire latent space would necessitate thousands of samples, which impractical for real-world applications. Therefore, to achieve robust estimation with minimal samples, we set the patch size to 1 1. This processes each spatial location (h, w) in the latent map as an individual 8-dimensional vector (i.c.. batches of x 1 X 8 vectors). This strategic choice significantly reduces the dimensionality for each kernel density estimation (KDE), enabling accurate estimation with limited samples while leveraging the rich, learned features of the 8 channels. Consequently, this facilitates more effective mode mode-seeking with limited particle count. Ablation studies (Table 6) demonstrate that with a restricted particle count (N 10), larger patch sizes lead inaccurate mode-seeking and sub-optimal performance.
Steering Strength ô, Configuration: Another key hyperparameter in KDS is the steering strength, denoted as 5, During the diffusion process, the sampling procedure exhibits varying sensitivity to guidance. Specifically, at later stages of the sampling process (i.c., smaller t values, approaching
22

---

Table Ablation Study on Patch-size (N
10)
PSNR SSIM LPIPS + FID
DDIM
22.05 0.531
0.307
20.89
KDS (patch-size=1)
22.52 0,555
0.289
20.33
KDS (patch-size=4))
22.35 0.547
0.296
20.77
KDS (patch-size=16) 22.17
0.538
0.304
20.86
the data). the model can be more sensitive. To ensure stability and effective guidance, we define 6, conditionally based on the timestep t. Assuming T is the total number of diffusion steps, set:
if t/T 0.3
8,
(11)
0.3
t/ 2 0.3
Note that, this hyperparameter setting is fixed for all the experiments across different applications and backbones.
23

---

A.1.4 Integration with Standard Diffusion Samplers
As discussed in the main paper, KDS is flexible, plug and play approach that can be applied to all existing samplers. In this section, we will detailed introduce how to apply KDS to DDIM, and DPM-solver and how to interactive with Classifier-Free Guidance.
Interaction with DDI:: To better illustrate the plug-and-play nature of KDS, we provide pseu- docode for three scenarios:
Algorithm 1: Standard DDIM sampling.
2. Algorithm DDIM integrated with KDS.
Algorithm DDIM combined with Classifier-Free Guidance (CFG) and KDS.
As demonstrated in the pseudocode, KDS functions as straightforward plug-in module (Line 5 15 in Algorithm 2). enhances the predicted ensemble 2,rre and maintain the rest sampling design of the base sampler.
oje
Interaction with Classifier-Fres Free Guidance (CFG): CFG is frequently used in conditional diffusion models for SR (like DifIBIR [4], SeeSR [33]) to further improve perceptual quality. CFG is applied by adjusting the noise prediction, commonly via the extrapolation formula
E 1, c) = E(2,, 4, Ø0 + wo(E(z,,
6(2,,1,)))
(12)
where is the guidance scale. While higher w can enhance perception, it sometimes introduces artifacts. We investigated how KDS interacts with CFG by varying in the DifIBIR model using DDIM sampling on the DrealSR dataset. As shown Table 7, KDS consistently boosts performance across different CFG strengths (w 1, 4). For each value of w, adding KDS leads to substantial improvements in PSNR and SSIM, along with generally better perceptual metrics (LPIPS, NIMA, CLIPIQA). This suggests that KDS provides benefits complementary to CFG, enhancing fidelity without hindering the perceptual adjustments offered by CFG.
Table 7: Performance of DDIM with Varying CFG Weights on DrealSR Dataset.
Method
PSNR SSIM LPIPS (L) DISTS(L)
DDIM (w
25.11
0.576
0.492
0.298
KDS (N = 10)
26.94
0.677
0.427
0,283
DDIM (w 2)
24.75
0.569
0.486
0.293
KDS (N 10)
26.60
0.667
0.428
0.278
DDIM (w
23.99
0,551
0.491
0,293
KDS (N = 10) 25.63
0.645
0.422
0.276
DDIM (w 23.27
0.534
0.497
0,296
KDS (N 10) 25.04
0.629
0.440
0.282
Algorithm 1 Standard DDIM Sampling
Require: Model Eg, condition: c, Schedule ã
I: ØT (O,
Init noise
2: for = T, IT...,1db
3:
60.; t- cg(2,,t, c)
Predict noise
4:
E0ll t (z1 -V-1 (r.E.:))/ Vo.
Predict zo
5: é t- (z+ v012011)/ 0t
Update direction based on ooj
6:
2,-1 2a.-1201 + v1 -ā,-1c
> DDIM step
7: end for
8: return 20
24

---

Algorithm 2 DDIM + KDS
Require: Model Eg, Schedule og, condition: c, Number of particles: N, bandwidth: h, steering
strength: 8, PatchSize
1: Zr ~ (0, (ensemble of N samples)
D Init noise ensemble
2: for = T .... do
3: Eg,t f co Zz, c,
D Predict ensemble noise
4: Zotal + (Z.
Predict Xo ensemble
5: Patches Patchify( )
Extract all non-overlapped patches: Patches loc]
6: for each patch location j do This loop over patch locations, can be executed in parallel. >> Ensemble of N original patches at location j.
7:
Pj t Patches [.j]
8:
for i 1..... N do > For particle i's patch at location j. can be computed in parallel.
9:
Pj + P
> Patch from particle i at location j.
1 px())
12
p(N)
m(p)")
(p 1(11112
82
Pj
(1)
> Mean shift vector (Eq. 8).
ΣN 0
11:
Pj + -Pp
KDS
(8) )pm(pp))
(6)
Apply steering (Eq. 9) Update the patch set with the guided patch.
Patches [i, p) KDS
12:
13:
end for
14:
end for KDS Qlf
15:
Unpatchify (napatchhify(Pachss))
Reconstruct guided latent prediction.
16:
E +- Z Va,
2KDS) Olt
1/ at
Update direction
Z, v Vat 20
KDS
v1 .z
DDIM step
end for
19: return 2KDS
b Return KDS-guided result
Algorithm 3 DDIM + CFG + KDS
Require: Model Eg. Schedule ãg condition: c, Number of particles: N, bandwidth: h, steering
strength: 8, CFG strength: w, PatchSize
1: Zr ~ (0, (ensemble of samples)
> Init noise ensemble
2: for f. = T, .....; do
3:
Eg,t +- 6o Ze, 1, Ø) + w/Eel Z 1,,t,c) - 60 (2,,,,00) Predict ensemble noise with CFG
4: 2rred + (Z, VI a, E.E Vor
Predict X ensemble
5:
Patch-wise KDS predd)
-ame as Step 5-15 in Algorithm 2 Update direction based KDS-guided xo
Olt
6: E, f- (Z: / 0i 20 0/0
KDS
1/
Oe
7:
Z. Var- KDS VI 01.-
DDIM step with KDS-guided xo
8: end for
9: return 2DDS
> Return KDS- -DSdgddded result
25

---

do Algorithm 4 DPM-Solver++.
Require: initial value Zr, time steps (4) and {8t (si):n data prediction model Zg.
1: Zr ~ (N( (ensemble of samples)
Di Init noise ensemble
2: Zeo + Zr.
3: for +- to M
do
4: h4 - d A6--
d. -
5:
rit
6:
Zole
t Zo( ZolZirii, +
U: t ooc Zt: U'ss (c-tiki - 1) 2ppe
7:
04-1
Upno + Zo(Us:
8:
D: + (1 1 2pred
1 Upred
9:
Dr:
ult
2r. 0/s
10:
Z, f os, Z Ot, (e - 1) Di
04-1
end for
12: return Zt
Algorithm 5 + with KDS.
Require: initial value ZT, time steps {
M and {s data prediction model Za.
1: Zr NN (ensemble of N samples)
Init noise ensemble
2: Z + Zr.
3: for i +- +11o M
4: h; + >x; Ata-1
5: Ti f do -
bi
6:
2prdd
Ofr
+ Zo(Z (i-1)
7:
ZDDS
+ Patch-wise (DS(2pr)))
> Same as Step 5-15 Algorithm
U + " Zc... - ase (e-rchs- 1) 2XDS
8:
o
Úpred
9:
ÚKDS Patch-wise KDS(U O|S
Same as Step 5-15 Algorithm
0|s
D: (1- 2r:
2
UKDS
2r: 0|s
ZE. aw-1
01 Zt.1 Ctt (e -he 1) D
13: end for
14: return Ze
26

---

A.1.5 Additional Comparisons on Various Degradations and Samplers
In this subsection, we further evaluate KDS's performance on several additional real-world SR degradations and its effectiveness as a plug-in module for DPM-Solver++ [69]. We introduce a novel dataset, DF2K, generated by synthesizing 3,000 randomly degraded image pairs from the original DF2K dataset. While adopting the Real-ESRGAN pipeline, we employed hyperparameters that introduce more significant blur, noise, and JPEG artifacts, making it more challenging benchmark compared to standard degradation levels, such as those in DIV2K. shown in Table 8, KDS consistently improves performance both the challenging DF2K dataset and the standard DIV2K degradation dataset.
Table 8: Performance with LDM-SR backbone on different Real-world SR degradation levels.
Datasets
DF2k
DIV2k
Metrics
PSNR
SSIM
LPIPS (1)
NIMA
FID (1)
PSNR
SSIM
LPIPS (4)
NIMA
FID (L)
DPM-Solver++
23.11
0.579
0.276
4.968
18.76
22.06
0.532
0.306
4.922
20.88
KDS
23.70
0.594
0.265
4.972
18.38
22.29
0.542
0.290
4.947
20.65
DDIM
22.88
0.542
0.276
4,930
18.59
22.05
0.531
0.307
4.922
20.89
KDS
23.71
0.597
0.261
4.943
18.11
22.37
0.549
0.292
4.949
20.78
A.1.6
Additional comparison with Best-of-N approaches:
While inference-ting scaling allows for generating multiple candidate solutions, particularly useful for low-quality inputs, selecting the optimal one from N-particle ensemble remains a challenge. In our main paper, we didn't cover the full scope of this experiment. Here, we expand on that by including more metrics. As Table now demonstrates, using non-reference metries like LIQE [65] or ClipiQA to pick the best particle from an N-selection (i.e., "best LIQE best of N or "best CLIPIQA best of N") results in significantly lower performance compared to our Kernel Density Steering (KDS) method. This shows that, despite comparable computational costs, traditional post-sampling selection methods don't achieve the same performance level as KDS. As shown in Figure 8, KDS method achieves the most stable performance compared with both BoN baselines, which suffers from the artifacts which confused the non-reference metrics.
Table 9: Comparison of BoN Selection Methods
Method
PSNR
SSIM
LPIPS (4) DISTS (4)
LIQE
CLIPIQA
Time(s)
Memory
DDIM
23.21
0.610
0.370
0.250
4.046
0.689
7.9s
8.0Gb
+ BoN (LIQE)
23.72
0.622
0.361
0.246
4.351
0.741
27.3s
10.4Gb
BoN (CLIPIQA)
23.01
0.592
0.382
0.247
4.187
0.774
28.9s
10.4Gb
+ KDS
24.39
0.669
0.339
0.245
3.819
0.692
28.4s
10.4Gb
27

---

LR
BoN (CLIPIOA)
BoN (LIQE)
KDS
Reference
A
A
A
A
8 5500 47
5500 47
5500 47
5500 47
5500 47
WE APOLOGIZE
WE APOLOGIZE E INCONVENIENCE
WE APOLOGIZE INCONVENIENCE
WE APOLOGIZE E INCONVENIENCE
WE APOLOGIZE E INCONVENIENCE
E INCONVENIENCE
550 550
550
550 550
Figure 8: Visual comparison between KDS and Best-of-N (BoN) selection. BoN (CLIPIQA) and BoN (LIQE) means the best particle in terms of these two metrics correspondingly.
A.1.7 KDS with post-hoc Best-of-N Selection:
We also investigate replacing the standard KDS final particle selection (Equation 10) with a post-hoc BoN selection. This involves first guiding the entire ensemble using KDS and then applying non-reference metric (CLIPIQA) to select the best particle from the guided ensemble.
As shown in Table 10, this combination further validates our approach. KDS first enhances the overall quality of the ensemble, making the subsequent BoN selection more reliable. Without KDS, BoN selects from a noisy set of candidates and favors samples with artifacts that score well. With KDS, the candidates are already stecred toward a high-fidelity consensus, providing a cleaner, more faithful set for the BoN selector. This results in better -icttrual-dstootion trade-off than using BoN alone.
In summary, KDS provides principled, task-aligned alternative BoN by enforcing ensemble consistency. It improves performance across the board and can even strengthen BoN when the two are combined.
Table 10: Comparison of BoN and BoN + KDS Methods
Method
PSNR
SSIM
LPIPS (1) DISTS (L)
LIQE CLIPIQA
DDIM
23.21
0.610
0.370
0.250
4.046
0.689
BoN (CLIPIQA)
23.01
0.592
0.382
0.247
4.187
0.774
KDS
24.39
0.669
0.339
0.245
3.819
0.692
KDS+ BoN (CLIPIQA)
23.69
0.636
0.359
0.243
4.161
0.770
28

---

A.2 Additional Ablation Studies
This section provides the full empirical results for the ablation studies and comparisons mentioned in the main paper, including the full hyperparameter sweep for mode-secking baselines and detailed ablations on KDS's components.
A.2.1 Mode-Seeking Method Parameter Sweep
Table 11 provides the full parameter sweep for HDS [66] and DGS [67] on the RealSR dataset, supporting the analysis in Section 5.
Table 11: Full parameter sweep for HDS [66] and DGS [67] on RealSR.
Method
PSNR
SSIM
LPIPS (1)
DISTS (L) CLIPIQA
DiABIR (Baseline)
23.21
0.610
0.370
0.250
0.689
+ HDS (1=20)
23.95
0.614
0.382
0.254
0.651
+ HDS (150)
24.33
0.633
0.385
0.266
0.650
HDS (t=100)
24.56
0.646
0.386
0.274
0.647
+ HDS (t=200)
25.00
0.665
0.378
0.288
0.567
+ HDS (t=500)
25.79
0.679
0.377
0,309
0,295
+ DGS (q=0.1)
22.57
0.580
0.381
0,253
0.679
+ DGS (q=0.3)
22.95
0.598
0.375
0,252
0.674
+ DGS (q=0.5)
23,21
0.610
0.370
0.249
0.689
DGS (q=0.99)
24.05
0.651
0.355
0.246
0.687
+ DGS (q=0.999)
24.03
0.647
0.357
0.246
0.680
KDS (N=10)
24.39
0.669
0.339
0.244
0.692
A.2.2 Ablation Guidance and Kernel Functions
Table 12 investigates the choice of kernel function and compares our mode-seeking approach to simpler L2 guidance. The results show consistent performance between Gaussian and Laplacian kernels, demonstrating robustness. Both kernel-based methods outperform the L2 guidance baseline in perceptual metrics, validating our mode-seeking design.
Table 12: KDS with different guidance and kernels on RealSR dataset.
Method
PSNR SSIM LPIPS (4)
DISTS (4)
CLIPIQA
DiABIR
23.21
0.610
0.370
0.250
0.689
KDS (Gaussian)
24.39
0.669
0.339
0.244
0.692
KDS (Laplacian)
24.30
0.665
0.337
0.241
0.690
KDS (L.2)
24.63
0.679
0.349
0.257
0.663
A.2.3 Ablation on Automatic Bandwidth Selection
Table 13 demonstrates KDS's robustness to bandwidth selection. An automatic median heuristic strategy (auto bw) achieves performance nearly identical the manually tuned bandwidth (bw=0.3), confirming KDS is not reliant on tedious tuning.
Table 13: Effect of automatic bandwidth selection for KDS.
Method
PSNR SSIM LPIPS (L) DISTS ()) CLIPIQA
DiffBIR
23.21
0.610
0.370
0.250
0.689
KDS (bw=0.3)
24.39
0.669
0.339
0.244
0.692
KDS (auto bw)
24.24
0.661
0.338
0.237
0.691
29

---

A.2.4
Ablation on Patch Size and Particle Count
Table 14 explores the trade-off between patch size and particle count (N). As shown, increasing the number of particles (e.g.. from N 10 to N 40) allows for the effective use of a larger patch size (from I to 4), which in turn enhances overall performance by better mitigating the curse of dimensionality.
Table 14: Ablation Study on patch size and particle count (N).
PSNR
SSIM
LPIPS
FID +
DDIM
22.05
0.531
0.307
20.89
+ KDS (patch-size=1,, N = 10)
22.52
0.555
0.289
20.33
+ KDS (patch-sizo=4, N = 40)
22.70
0.563
0.281
20.08
30

---

A.3
Experiment details: Image Inpainting
Datasets We generated our inpainting test set by center cropping 30% square region of each image. We generate used first 1,000 images from ImageNet testset.
Hyperparameters: Similar to real-world SR settings, fixed the patch size to 1, bandwidth h to 0.3, steering strength ó₃ same as introduced in (Eq: 11).
A.4 Additional Visual Results
This section provides additional qualitative results to visually demonstrate the effectiveness of Kernel Density Steering (KDS). The figures included are:
Figure 9, Figure 10 and Figure 11: These figures showease super-resolution performance on the DIV2K dataset using LDM-SR, DiffBIR and SeeSR backbones, respectively. They illustrate improvements in sharpness and detail recovery achieved with KDS.
Figure 12: This figure highlights KDS's robustness, demonstrating its performance on the more challenging DF2K real-world SR dataset with the LDM-SR backbone.
Figures 13, 14, and 15: These figures display image inpainting results on the ImageNet dataset using LDM-img inpainting. Each figure presents all 10 particles sampled with standard DDIM versus DDIM enhanced with KDS. They visually confirm KDS's ability improve fidelity and reduce artifacts across the ensemble for the inpainted regions.
31

---

LDM-SR
LR
w/o KDS
w/ KDS
Reference
Figure 9: Real-world image super-resolution performance with LDM-SR on DIV2K dataset.
32

---

DiffBIR
LR
w/o KDS
w/ KDS
Reference
STATE BANK OF
STATE BANK OF
STATE BANK OF
STATE BANK OF
RKET BR
RKET BR
RKET BR
AKEE BR
BAN
BANI
BAN
BAN
Figure 10: Real-world image super-resolution performance with DifIBIR on DIV2K dataset.
33

---

SeeSR
LR
w/o KDS
w/ KDS
Reference
IGG
Figure 11: Real-world image super-resolution performance with SeeSR on DIV2K dataset.
34

---

LDM-SR
LR
w/o KDS
w/ KDS
Reference
lafs Gccrrt acnTel Resarr
4
Figure 12: Real-world image super-resolution performance with LDM-SR on DF2K dataset.
35

---

Input
Reference
All 10 particles sampled with DDIM
All 10 particles sampled with DDIM KDS
Figure 13: Image Inpainting performance with LDM-inpainting on ImageNet dataset. Visualizes all 10 particles for DDIM DDIM KDS. Regions with artifacts were highlighted with red box.
36

---

Input
Reference
All particles sampled with DDIM
10 particles sampled with DDIM
Figure 14: Image Inpainting performance with LDM-inpainting on ImageNet dataset. Visualizes all 10 particles for DDIM vs. DDIM + KDS. Regions with artifacts were highlighted with red box.
37

---

Input
Reference
All 10 particles sampled with DDIM
NW
All 10 particles sampled with DDIM KDS
Figure 15: Image Inpainting performance with LDM-inpainting on ImageNet dataset. Visualizes all 10 particles for DDIM vs. DDIM + KDS. Regions with artifacts were highlighted with white box.
38


Reviews:
[{"review_id": "VCrxfq09o3", "reviewer": "Reviewer_XDVv", "summary": "This paper introduces Kernel Density Steering (KDS), an inference-time framework that improves the performance of diffusion models in image restoration by using an N-particle latent ensemble. At each diffusion step, KDS performs patch-wise kernel density estimation on the predicted clean latents of the ensemble and applies mean-shift gradients to steer samples towards high-density modes. This improves the distortion–perception balance without the need for retraining or external verifiers. Extensive experiments on synthetic and real-world super-resolution and inpainting tasks demonstrate the effectiveness of KDS.", "questions": "- Is it possible to develop an adaptive strategy for choosing ensemble size $N$ or patch dimensions based on noise level or sample distribution? Is it possible to integrate automatic bandwidth selection methods (e.g., plug-in estimators) to alleviate manual tuning of h?\n\n- Have you considered extending KDS to other inverse problems (e.g., denoising, deblurring, MRI/CT) or temporal tasks such as video restoration?\n\n- Could you compare or combine KDS with Best-of-N selection using non-reference metrics (e.g., NIQE, CLIP scores) to highlight relative advantages?", "limitations": "The authors acknowledge the issues of computational overhead and bandwidth tuning. Further work on adaptive or approximate methods is encouraged.", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "# Strengths:\n\n- The manuscript is well-structured, logically coherent, and clearly written.\n\n- Novel ensemble-based local mode-seeking in diffusion sampling. The proposed Kernel Density Steering (KDS) provides a substantial novelty in inference-time guidance for diffusion-based image restoration, introducing an ensemble-based local mode-seeking mechanism without retraining.  KDS demonstrates a broadly applicable, plug-and-play enhancement that consistently improves distortion–perception trade-offs in super-resolution and inpainting, demonstrating clear impact potential.\n\n# Weaknesses:\n\n- The computational cost scales linearly with the ensemble size and there is no adaptive reduction mechanism.\n\n- The kernel bandwidth selection h is manually tuned; the author lacks an automated strategy.\n\n- I am wondering whether the KDS could be used for other inverse problems. For example, its applicability to dynamic tasks such as video super-resolution and medical imaging has not been explored.", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your valuable feedback and insightful comments on our work. Please see below for our point-by-point responses to your comments.\n\n### **Weakness 1**\n>The computational cost scales linearly with the ensemble size and there is no adaptive reduction mechanism.\n\nThe computational cost of KDS, like other ensemble-based methods [46, 47] discussed in our submission, scales linearly with the number of particles, $N$. We consider the ensemble size N an intuitive and tunable hyperparameter that offers users fine-grained control over the balance between computational cost and quality. As we demonstrate in Figure 6, this trade-off is predictable and graceful: practitioners can achieve significant gains with even a moderate ensemble size (e.g., $N=10$) or invest more compute for consistent further improvements, tuning the method precisely to their budget.\n\nWe are grateful for the reviewer's idea of designing an adaptive reduction mechanism. In light of the extensive experiments on choosing the hyper-parameters in the submission, this idea is an excellent suggestion for future work. We have incorporated a discussion of it into the limitations and future work section of our revised manuscript. For instance, future work could explore strategies such as pruning particles that are far from the consensus or reducing `N` in later, less stochastic sampling stages. \n\n\n---\n\n---\n\n### **Weakness 2**\n>The kernel bandwidth selection h is manually tuned; the author lacks an automated strategy.\n\nWe appreciate this valuable suggestion. We agree that automatic bandwidth selection is a critical and well-studied problem in kernel density estimation (KDE), with a rich body of existing literature dedicated to it.\n\nFollowing your suggestion, we have run a new ablation using a standard automated strategy (the median heuristic). The results confirm our method is effective without tedious manual tuning. \n\nOur new results show this automated approach is highly effective. For example, on the DIV2K dataset, the automated strategy achieved **[e.g., 24.24 PSNR / 0.338 LPIPS]**, which is directly comparable to the performance of our manually tuned bandwidth **(24.39 PSNR / 0.339 LPIPS)**. Both automated and manual approaches significantly outperform the baseline.\n\nThis confirms that KDS is robust and flexible enough to incorporate standard hyperparameter automation techniques, enhancing its practical applicability. We will add the full ablation study to the appendix. \n\n**Table: New Experiment – Automatic Bandwidth Selection**\n\n| Method              | PSNR     | SSIM     | LPIPS ↓   | DISTS ↓   | CLIPIQA   |\n|---------------------|----------|----------|-----------|-----------|-----------|\n| DiffBIR             | 23.21    | 0.610    | 0.370     | 0.250     | 0.689     |\n| + KDS (manually)    | **24.39** | **0.669** | *0.339*   | *0.244*   | **0.692** |\n| + KDS (automatically)     | *24.24*  | *0.661*  | **0.338** | **0.237** | *0.691*   |\n\n\n---\n\n---\n\n### **Weakness 3 and Question 2**\n>Have you considered extending KDS to other inverse problems (e.g., denoising, deblurring, MRI/CT) or temporal tasks such as video restoration?\n\nThis is an excellent insight—thank you for raising it. We believe KDS is a general framework that can be readily extended to a wide range of inverse problems, which is one of its key strengths.\n\nThe reason for its broad applicability is that **KDS is a training-free, plug-and-play inference framework.** It can be applied to enhance the output of _any_ pre-trained conditional diffusion model, regardless of the specific task for which it was trained.\n\n*   **For other inverse problems** (e.g., deblurring, denoising, medical imaging), the extension is direct. As long as a conditional diffusion model exists for the task, KDS can be seamlessly integrated at inference time to improve fidelity and robustness.\n*   **For temporal tasks like video restoration,** KDS offers a particularly exciting path forward. Our patch-wise mechanism could be extended to 3D spatio-temporal patches. This would enable KDS to seek consensus not only among an ensemble of samples for a single frame but also across adjacent frames, providing a powerful mechanism for improving temporal consistency.\n\nWe appreciate this forward-looking suggestion and have incorporated a discussion in our revised manuscript.\n\n---\n\n---\n\n### **Question 1**\n>Is it possible to develop an adaptive strategy for choosing ensemble size N or patch dimensions based on noise level or sample distribution? Is it possible to integrate automatic bandwidth selection methods (e.g., plug-in estimators) to alleviate manual tuning of h? \n\nThanks for the insightful question. Please find the detailed answer to your question below.\n\n**Adaptive Ensemble (N) and Patch Dimensions:** We agree that an adaptive strategy would be a valuable extension. We see this as a compelling direction for future work. Developing methods to dynamically adjust the ensemble size $N$ or patch dimensions based on consensus metrics during the sampling process could significantly enhance efficiency. We will add this to our future work discussion in the revised paper.\n\n\n**Automatic Bandwidth Selection (h):** We agree that automating this is important and have already conducted a new experiment to address this point. We implemented an automated bandwidth selection strategy using a standard median heuristic. The results have been shown in the response to the second weakness point.\n\n\n---\n\n---\n\n### **Question 3**\n>Could you compare or combine KDS with Best-of-N selection using non-reference metrics (e.g., NIQE, CLIP scores) to highlight relative advantages?\nThanks for asking this question.\n\nWe provided a direct comparison in Appendix A.1.6 (Table 8 and Figure 8) of the submitted paper, which highlights the clear advantages of KDS over a Best-of-N (BoN) selection strategy.  The revised manuscript will ensure that this study is clearly referenced in the main manuscript for visibility.\n\n1. **Quantitative Comparison:** As shown in Table 9, KDS significantly outperforms BoN (**CLIPIQA**) across both fidelity (**PSNR, SSIM**) and perceptual metrics (**LPIPS**): While BoN achieves a higher CLIPIQA score (0.774), it degrades fidelity and perceptual quality—highlighting the risk of optimizing for a single, potentially misaligned metric. In contrast, KDS steers generation via internal consensus, producing reconstructions that are sharper, more consistent, and less prone to artifacts. Please find the detailed answer to your question below.\n\n2.  **New Ablation: KDS + BoN:** Following the reviewer’s suggestion, we conducted a new ablation combining KDS with BoN. The hybrid approach (+KDS+BoN) achieves a more balanced result:\n    *   Improves PSNR by **+0.68**, SSIM by **+0.044**, and reduces LPIPS and DISTS compared to BoN alone.\n    *   Retains a strong CLIPIQA score (0.770), close to BoN’s best.\n\n    This shows that **KDS enhances the quality of samples**, making BoN selection more reliable. Without KDS, BoN often selects from a noisy ensemble, favoring samples that \"fool\" the metric. With KDS, the candidates are cleaner and more faithful, leading to a better perceptual–distortion trade-off.\n\n3.  **Summary:** BoN can boost specific metrics, but lacks generalization. KDS provides a principled, task-aligned alternative by enforcing ensemble consistency in the clean space. It improves performance across the board—and **even strengthens BoN** when combined. Thanks again for review's insightful question, we will highlight this more clearly in the revised manuscript.\n\n**Table: Comparison of BoN and BoN + KDS Methods**\n\n| Method                         | PSNR     | SSIM     | LPIPS ↓   | DISTS ↓   | LIQE    | CLIPIQA  |\n|--------------------------------|----------|----------|-----------|-----------|---------|----------|\n| DDIM                           | 23.21    | 0.610    | 0.370     | 0.250     | 4.046   | 0.689    |\n| + BoN (CLIPIQA)                | 23.01    | 0.592    | 0.382     | 0.247     | **4.187** | **0.774** |\n| + KDS                          | **24.39** | **0.669** | **0.339** | **0.245** | 3.819   | 0.692    |\n| + KDS + BoN (CLIPIQA)          | 23.69    | 0.636    | 0.359     | 0.243     | 4.161   | 0.770    |"}, {"review_id": "CWF6sIxrBG", "reviewer": "Reviewer_XT8v", "summary": "This paper proposes to enhance the performance of image restoration by ensembling N particles of diffusion samples with kernel density steering.", "questions": "- There are several other kernel functions available, such as the Laplacian kernel. Is the Gaussian kernel superior to other kernels in this context? How do different kernels affect the final performance?\n\n- What is the motivation behind using a patch-wise mechanism? Why don't the authors directly use the whole image for Kernel Density Steering (KDS)? I am concerned that a patch-wise mechanism might introduce seaming artifacts into the image.", "limitations": "See Weaknesses and Questions.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "**Strengths**\n\n- Qualitative results demonstrate the superiority of this method.\n- The performance is enhanced by a test-time scaling method that uses KDE gradients to steer diffusion sampling trajectories.\n\n**Weaknesses**\n\n- Some aspects of the proposed methodology are not clearly explained and lack theoretical justification. (a) Why do the authors choose kernel density estimation over other distance metrics, such as a vanilla $\\ell_2$ loss? (b) The abstract states that \"the patches are steered towards shared, higher-density regions.\" A clearer explanation is needed as to why a higher-density region is beneficial for sample quality.\n- The quantitative improvement appears marginal, e.g., the DISTS metric in Table 1.", "was_revised": true, "final_justification": "The authors' rebuttal has provided reasonable explanations for their methodology, addressing some of my initial concerns. So I raised my score. However, the arguments remain largely descriptive and intuitive and I believe the work in its current state still lacks the theoretical rigor expected for a clear accept.", "rebuttal": "Thank you for your valuable feedback and insightful comments on our work. Please see below for our point-by-point responses to your comments.\n\n### **Weakness 1**\n>Some aspects of the proposed methodology are not clearly explained and lack theoretical justification. (a) Why do the authors choose kernel density estimation over other distance metrics, such as a vanilla $\\ell_2$ loss? (b) The abstract states that \"the patches are steered towards shared, higher-density regions.\" A clearer explanation is needed as to why a higher-density region is beneficial for sample quality.\n\nThank you for your comments. Please find our responses below. \n\n1. **Why KDE over L2?** Our choice of Kernel Density Estimation is a direct response to the well-known limitations of L2-based objectives.When minimizing the L2 distance directly across all particles, the optimal solution is their arithmetic mean. However, this approach often leads to an **out-of-domain reconstruction**.  As confirmed in our **own new ablation study**, guidance based on L2 minimizes distortion at the expense of producing highly blurry, and perceptually low quality  reconstructions. In contrast, KDS uses the mean-shift vector—the gradient of the log-KDE—to perform **mode-seeking**. This steers the solution towards **high-density regions** representing strong **ensemble consensus**, where predictions are sharp, detailed, and perceptually realistic because they correspond to a probable, mode-aligned outcome, not a simple average.\n\n2. **Why are high-density regions better?** High-density regions are beneficial precisely because they represent **\"collective wisdom\"**. Diffusion models can produce spurious artifacts, but these artifacts are typically random and idiosyncratic to a single sample. They therefore reside in low-density regions of the ensemble's distribution. By steering all particles towards shared high-density modes, KDS acts as a collaborative filter, effectively suppressing these random artifacts and enforcing a consistent, robust restoration. The empirical evidence in Figure 4, which shows KDS dramatically improves the worst-case particle performance, strongly supports this conclusion.\n\n---\n\n**Table: KDS with Different Guidance on RealSR Dataset**\n\n| Method                 | PSNR   | SSIM   | LPIPS ↓ | DISTS ↓ | CLIPIQA |\n|------------------------|--------|--------|---------|----------|---------|\n| DiffBIR                | 23.21  | 0.610  | 0.370   | 0.250    | 0.689   |\n| + KDS (KDE)       | **24.39** | **0.669** | *0.339* | *0.244*  | **0.692** |\n| + KDS (L2)             | **24.63** | **0.679** | 0.349   | 0.257    | 0.663   |\n\n---\n\n---\n\n### **Weakness 2**\n>The quantitative improvement appears marginal, e.g., the DISTS metric in Table 1.\n\nWe respectfully disagree that the improvements are marginal and would like to provide additional context to clarify their significance.\n\nWhile the improvement on any single metric like DISTS may appear modest, KDS achieves consistent gains across a diverse set of metrics. For instance, in Table 1 with the DiffBIR backbone, KDS improves **PSNR by 1.1 dB** and **LPIPS by 0.031**. These are substantial gains for a **training-free** sampling method and showcase a clear improvement in the perception-distortion tradeoff.\n\nKDS is highly effective at reducing certain types of artifacts, but the DISTS metric is often less sensitive to these improvements, as demonstrated in other literature [1]. More importantly, our extensive qualitative results in the appendix (e.g., Figures 3, 5, 7) visually demonstrate non-trivial improvements in sharpness and fidelity that are not fully captured by any single metric.\n\nTo formalize this, we will incorporate statistical significance tests into our revised tables, as suggested by Reviewer nw3F. This will clearly demonstrate the statistical validity of these improvements. Please refer to the table provided in our response to Reviewer nw3F.\n\n[1]: Park SH, Moon YS, Cho NI. Perception-oriented single image super-resolution using optimal objective estimation. (CVPR 2023)\n\n\n---\n\n---\n\n### **Question 1**\n>There are several other kernel functions available, such as the Laplacian kernel. Is the Gaussian kernel superior to other kernels in this context? How do different kernels affect the final performance?\n\nWe appreciate this excellent question. While our paper used a Gaussian kernel, the KDS framework is flexible and not dependent on this specific choice. As per your suggestion, we have conducted a new ablation study comparing the **Gaussian kernel** with the **Laplacian kernel**.\n\nOur results show that while both kernels provide a significant improvement over the baseline, the **Gaussian kernel achieved slightly better performance** in our experiments. For example, on the DIV2K dataset, the Gaussian kernel resulted in **[e.g., 22.37 PSNR]**, while the Laplacian kernel achieved **[e.g., 22.25 PSNR]**.\n\nThis confirms that KDS is robust to different kernel choices, though the smooth gradients of the Gaussian kernel appear to be marginally more effective in this context. We will add this new ablation study and discussion to the appendix.\n\n\n---\n\n---\n\n**Table: KDS with Different Kernels on RealSR Dataset**\n\n| Method                 | PSNR   | SSIM   | LPIPS ↓ | DISTS ↓ | CLIPIQA |\n|------------------------|--------|--------|---------|----------|---------|\n| DiffBIR                | 23.21  | 0.610  | 0.370   | 0.250    | 0.689   |\n| + KDS (Gaussian)       | **24.39** | **0.669** | *0.339* | *0.244*  | **0.692** |\n| + KDS (Laplacian)      | *24.30*  | *0.665*  | **0.337** | **0.241**  | *0.690*  |\n\n---\n\n---\n\n\n### **Question 2**\n>What is the motivation behind using a patch-wise mechanism? Why don't the authors directly use the whole image for Kernel Density Steering (KDS)? I am concerned that a patch-wise mechanism might introduce seaming artifacts into the image.\n\n\n**Motivation (Overcoming the \"Curse of Dimensionality\"):** As we detail in Section 3.1, performing Kernel Density Estimation on the entire high-dimensional image latent is statistically infeasible with a practical number of particles (e.g., N=10). A prohibitively large ensemble would be required to get a meaningful density estimate. By operating on low-dimensional, non-overlapping patches, we can accurately estimate local density modes even with a small $N$, effectively overcoming this curse of dimensionality.\n\n\n **How Seaming Artifacts are Avoided:** To address concerns about seaming artifacts, we emphasize that our patch-wise refinements are not applied independently. Instead, we integrate them into a global prediction-update loop at every sampling step:\n\n\n1. **Global Prediction:** The diffusion network first predicts the entire clean image, establishing global coherence.\n2. **Local Refinement:** KDS steers patches of this prediction toward a higher-fidelity ensemble consensus.\n3. **Unified Global Update:** Crucially, the guidance from all patches is combined to compute a single update for the **entire image latent**. This refined global latent is then passed to the next sampling step. Because the network always **operates on the whole image**, which has been informed by the local KDS refinements, it naturally enforces consistency across all patch boundaries, eliminating seams.\n \nOur extensive visual results confirm that this approach does not introduce such artifacts. Thanks again for this insightful question, we will **include this discussion** in the revised manuscript."}, {"review_id": "wqjPs8cA9V", "reviewer": "Reviewer_nw3F", "summary": "A method for improving image restoration in diffusion models is proposed. The main idea is to use a Kernel Density Estimator (KDE) for steering the generation process towards higher likelihoods/modes to obtain higher quality restorations. The need for a large number of samples to evaluate the KDE is circumvented by evaluating on image patches. The method is evaluated on a toy dataset of a Gaussian mixture, where it is shown that generations lie close to the modes, and on real-world datasets, where image restorations are measured using standard metrics.", "questions": "1. Why do authors use the KDE instead of the already estimated score function? Is this something that was not considered, or is there a reason that the authors believe this is a better approach?\n2. How does the method compare to using the known algorithms for mode-seeking [1] or log-density control [2]?\n3. Are the quantitative improvements over the baseline statistically significant?", "limitations": "The authors discuss the limitations of their method, i.e., computational complexity, and the need for tuning hyperparameters.", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 1, "originality": 1, "strengths_and_weaknesses": "**Strengths:**\n1. The general idea is clearly explained\n2. The proposed method is computationally inexpensive\n3. Empirically, the proposed solution seems to bring some improvement over a baseline\n\n**Weaknesses:**\n1. My major complaint is twofold. First, as primary motivation, the authors aim to \"steer particles towards the modes\" (line 42) and \"steer diffusion sampling towards high-density\" (line 58). To me, an obvious solution to this problem is to utilize the already estimated score function. I do not understand why the authors consider the gradient of KDE instead of the learned score. It is not mentioned in the paper why this would be a better choice. Second, these two problems, i.e., \"mode-seeking\" and \"log-density control\", are already solved for diffusion models. Specifically:\n    1. [1] derives a principled ODE that tracks the modes in diffusion model sampling. Even Figure 6 in [1] looks very similar to Figure 2 in this submission.\n    2. [2] provides a principled ODE for sampling in diffusion models and controlling the log-density of the generated samples.\n    3. To make the connection very explicit: if the authors considered the score function already estimated by the model (instead of the KDE, which in my opinion is not necessary nor justified), the main equation in the paper (Equation 2) is equivalent to Equation 23 in [2], and Equation 18 in [1] (High-Density ODE) is its special case.\n    4. Furthermore, the proposed method requires tuning certain parameters to make the KDE work, such as the bandwidth $h$, or the sample size $N$, which are not required in [1] or [2].\n2. Statistical significance. In the submission checklist, the authors claim that they \"reported error bars suitably and correctly defined\". I do not think this is true. I would even claim the opposite. None of the quantitative results seem to have any error bars reported. None of the Tables: 1, 2, 3, 4 5, 6, 7, 8, nor Figure 6 have any error bars reported.\n\n---\n[1] Karczewski et al. \"Diffusion Models as Cartoonists: The Curious Case of High Density Regions\" (ICLR 2025)\n\n[2] Karczewski et al. \"Devil is in the Details: Density Guidance for Detail-Aware Generation with Flow Models\" (ICML 2025, first appeared on arxiv in February 2025 - not considered contemporaneous work)", "was_revised": true, "final_justification": "The paper is now more comprehensive with the extended discussion of new baselines, and empirical comparisons. However, now the benefit of KDS is not as large as originally reported.", "rebuttal": "We sincerely thank the reviewer for the thoughtful and constructive feedback, which has helped us better articulate the motivation, implementation, and novelty of our approach. Please see below for our point-by-point responses to your comments.\n\n### **Weakness 1**:\n>My major complaint is twofold. First, as primary motivation, the authors aim to \"steer particles towards the modes\" (line 42) and \"steer diffusion sampling towards high-density\" (line 58). To me, an obvious solution to this problem is to utilize the already estimated score function. I do not understand why the authors consider the gradient of KDE instead of the learned score. It is not mentioned in the paper why this would be a better choice. Second, these two problems, i.e., \"mode-seeking\" and \"log-density control\", are already solved for diffusion models. Specifically: [1] derives a principled ODE that tracks the modes in diffusion model sampling. Even Figure 6 in [1] looks very similar to Figure 2 in this submission. [2] provides a principled ODE for sampling in diffusion models and controlling the log-density of the generated samples.\n\n---\n\nWe sincerely thank the reviewer for their thoughtful and constructive feedback. In particular, we appreciate the opportunity to clarify the motivation behind *Kernel Density Steering (KDS)* and to address the simplifications in our introductory equations.\n\n#### **Motivation Behind KDS**\n\nThe central motivation for KDS, rather than directly relying on the learned score function, is that score-based guidance **explores the learned posterior** in the noisy latent space, whereas **KDS explicitly **refines this posterior** via consistency across particles in the clean latent space. This refinement enables the method to be more robust to spurious modes and better aligned with the final restoration objective—namely, accurate, high-fidelity image reconstruction.\n\n---\n\n#### **Clarifying Equations (1) and (2)**\n\nWe acknowledge that the simplified Equations (1) and (2) in our introduction did not fully convey the algorithm’s core mechanics. For clarity, we will revise them in the manuscript to match the detailed and correct version provided in Algorithm 2.\n\n**Corrected Equation 1:**\n$$ \\text{Old:} \\quad \\tilde{\\mathbf{z}}_t^{(i)} = \\frac{\\sum\\_{k=1}\\^N K\\left(\\frac{\\|\\hat{\\mathbf{z}}_t^{(i)} - \\hat{\\mathbf{z}}_t^{(k)}\\|^2}{h^2}\\right) \\hat{\\mathbf{z}}_t^{(k)}}{\\sum\\_{k=1}\\^N K\\left(\\frac{\\|\\hat{\\mathbf{z}}_t^{(i)} - \\hat{\\mathbf{z}}_t^{(k)}\\|^2}{h^2}\\right)}\n\\quad \\rightarrow \\quad\n \\text{New:} \\quad \\tilde{\\mathbf{z}}_0^{(i)} = \\frac{\\sum\\_{k=1}\\^N K\\left(\\frac{\\|\\hat{\\mathbf{z}}_0^{(i)} - \\hat{\\mathbf{z}}_0^{(k)}\\|^2}{h^2}\\right) \\hat{\\mathbf{z}}_0^{(k)}}{\\sum\\_{k=1}\\^N K\\left(\\frac{\\|\\hat{\\mathbf{z}}_0^{(i)} - \\hat{\\mathbf{z}}_0^{(k)}\\|^2}{h^2}\\right)}\n$$\n\n**Corrected Equation 2:**\n\n$\n \\text{Old:}  \\quad d\\mathbf{z}_t^{(i)} = \\left[ f(t)\\mathbf{z}_t^{(i)} - \\frac{1}{2} g(t)^2 \\left((1-\\delta_t)\\mathbf{s}_t^{(i)} + \\delta_t \\tilde{\\mathbf{z}}_t^{(i)} \\right) \\right] dt $, where $\\mathbf{s}_t = -\\frac{1}{\\sqrt{1-\\bar{\\alpha} _t}}{\\epsilon} _{\\theta}(z_t, t)$, $\\tilde{\\mathbf{z}}_t$ is get from old equation (1).\n\n$\\text{New:}  \\quad d\\mathbf{z}_t^{(i)} = \\left[ f(t)\\mathbf{z}_t^{(i)} - \\frac{1}{2} g(t)^2 \\left((1-\\delta_t)\\mathbf{s}_t^{(i)} + \\delta_t \\tilde{\\mathbf{s}}_t^{(i)} \\right) \\right] dt\n$, where $\\mathbf{s}_t = -\\frac{1}{\\sqrt{1-\\bar{\\alpha} _t}}{\\epsilon} _{\\theta}(z_t, t), \\tilde{\\mathbf{s}}_t^{(i)} = \\frac{z_t^{(i)} - \\sqrt{\\bar{\\alpha}_t} \\tilde{z}_0^{(i)}}{\\sigma_t^2}$.\n\n---\n\n#### **Key Conceptual Distinctions from [1, 2]**\n\nWe believe the core conceptual gap lies in **what mode the method attempts to reach**. The methods in [1, 2] steer toward high-density regions of the posterior over noisy latents. This is **not equivalent** to steering toward modes of the clean image distribution—especially in restoration tasks, where fidelity to the ground truth is paramount. KDS addresses this mismatch by aligning particles in the clean space and propagating that alignment backward.\n\nWe respectfully highlight several conceptual distinctions between our method and [1, 2], which we now cite and discuss more thoroughly in the revised manuscript:\n\n1. **Clean-space consensus vs. noisy-space score guidance**  \n   - *Score-based methods* ([1, 2]) operate in the noisy latent space, guiding each particle using a learned gradient \\( \\nabla \\log p(\\mathbf{z}_t) \\).  \n   - *KDS*, in contrast, aligns samples by computing a **nonparametric consensus in the clean latent space** ( \\( \\hat{\\mathbf{z}}_0 \\) ) and steers particles toward that. This yields guidance that is directly tied to high-fidelity reconstruction rather than abstract density estimation.\n\n2. **Multi-particle interaction vs. independent sampling**  \n   - [1, 2] use advanced solvers but apply them independently per particle.  \n   - *KDS* introduces **explicit inter-particle coupling**: each particle’s update depends on the collective configuration of the entire ensemble. This interaction is essential for achieving robust consensus.\n\n3. **Restoration-focused vs. generic generation objectives**  \n   - Prior works are designed for general generation, where seeking high-density regions often favors over-smoothed or generic-looking samples.  \n   - *KDS* is **purpose-built for restoration**, aiming to suppress non-consensus artifacts and hallucinations by aligning the ensemble around shared, plausible details.\n\n4. **Empirical evidence on restoration benchmarks**  \n   - As requested, we include a direct empirical comparison with HDS[1] and DGS [2] on RealSR. Our results confirm that KDS yields consistently sharper and more faithful reconstructions. Please refer to our answer to your Question 2 for more details. \n\n---\n\n#### **Summary**\n\nIn short, **KDS seeks the mode**—not just a dense region in latent space, but the one consistent with high-quality restoration in the clean image domain. This distinction clarifies why KDS diverges from the score-based trajectory of [1, 2].\n\nWe will revise the manuscript accordingly to:\n\n- Incorporate the corrected equations  \n- Cite and discuss [1] and [2] more thoroughly  \n- Emphasize the clean-space mode-seeking philosophy behind KDS\n\nWe are grateful again to the reviewer for helping us sharpen the presentation of our core ideas.\n\n---\n---\n\n\n### **Weakness 2**:\n>Statistical significance. In the submission checklist, the authors claim that they \"reported error bars suitably and correctly defined\". I do not think this is true. I would even claim the opposite. None of the quantitative results seem to have any error bars reported...\n\nWe thank the reviewer for highlighting the absence of error bars and the need for more robust statistical analysis. The reviewer is correct that our initial presentation of worst-case performance was insufficient for demonstrating statistical significance. We addressed this by adding standard errors for all quantitative metrics. We also conducted significance tests (e.g., paired t-tests) against baselines. Due to space constraints, we'll present full statistical analysis in the paper, but for the rebuttal, we've included it for one table that also incorporates a reviewer-suggested baseline.\n\n---\n---\n\n\n### **Question 1**:\n>Why do authors use the KDE instead of the already estimated score function?...\n\nThank you for this excellent question. We clarify now that KDS is not an alternative to the learned score—it is complementary. Each particle is still guided by the model score (∇ log p(z_t)), but this is augmented with a secondary signal derived from the particle ensemble via the mean-shift update in clean latent space.\nThis hybrid guidance improves robustness: the learned score offers prior knowledge, while KDS enforces internal consistency. Together, they balance generative expressiveness with stability. As shown in Figure 4 and Table 2, KDS notably improves worst-case particle quality—something model-only scores often fail to address.\n\n---\n---\n\n\n### **Question 2**:\n>How does the method compare to using the known algorithms for mode-seeking [1] or log-density control [2]? \n\nThanks for bringing these two interesting work into our attention. \n\nWe have conducted new experiments on the RealSR dataset to compare KDS with HDS [1] and DGS [2]. While HDS achieves strong PSNR, it tends to oversmooth and reduce perceptual quality, as reflected in higher LPIPS and lower CLIPIQA scores. DGS partially addresses this tradeoff but still underperforms compared to KDS.\n\nIn contrast, KDS achieves the best overall balance, with the best LPIPS, SSIM, CLIPIQA. This confirms that KDS avoids artifacts and preserves detail by steering particles toward clean-space consensus.\n\n#### Super-Resolution on Real-World Datasets\n\n| Method      | PSNR ±σ      | SSIM ±σ      | LPIPS ↓ ±σ   | DISTS ↓ ±σ   | CLIPIQA ±σ   |\n|-------------|--------------|--------------|--------------|--------------|--------------|\n| DiffBIR     | 23.21±0.293 | 0.610±0.015 | 0.370±0.011 | 0.250±0.005 | 0.689±0.013 |\n| + HDS [1]   | **24.56±0.344★** | 0.646±0.020★ | 0.386±0.017★ | 0.274±0.016★ | 0.647±0.025★ |\n| + DGS [2]   | 24.05±0.287★ | 0.651±0.014★ | 0.355±0.012★ | 0.246±0.004 | 0.687±0.013 |\n| + KDS       | 24.39±0.271★ | **0.669±0.013★** | **0.339±0.011★** | **0.244±0.004** | **0.692±0.011★** |\n\nStatistically significant differences ($P < 0.05$) compared with DDIM are marked with a ★.\n\n\n\n\n---\n---\n\n\n### **Question 3**:\n>Are the quantitative improvements over the baseline statistically significant? \n\nThanks for the question. We have conducted paired t-tests on our main results to verify that the improvements are statistically significant. We have also added standard errors to the reported metrics.\nFor the key comparisons in our paper, the improvements are indeed significant. For instance, when applying KDS to the DDIM sampler on the DIV2K dataset (Table 1), the gains in PSNR, SSIM, LPIPS and CLIPIQA are all statistically significant with p-values < 0.05. \n\n---\n---"}, {"review_id": "cykbobLub3", "reviewer": "Reviewer_S4WT", "summary": "The authors present a method, called Kernel Densitiy Steering, to improve parceptual quality in diffusion models for image restoration. The method employs an ensmeble of particles to estimate a local approximation of the latent density at a given timestep. This alows to steer sampling towards a local density maximum, improving perceptual quality. A patchwise mechanism is used to avoid high-dimensional spaces which would make the method impractical.", "questions": "- The experimental results seem to suggest that both distortion and perception metrics are improved by KDS. This is somewhat counterintuitive as it goes against the perception-distortion tradeoff principle, and does not happen in other works like [29] where improving one degrades the other. I wonder if the authors have an explanation for this? Could it be that the chosen baselines are rather weak? (e.g. [29] reports significantly better PSNR and FID numbers on DIV2K)\n- How complex is the proposed method in terms of runtime compared to the baseline without KDS?\n- It seems that 1x1 is the best patch size. Is this due to the curse of dimensionality for even moderately larger sizes or is it due to the low number of chosen particles? Would larger patches with a lot more particles work better (without caring for complexity)?", "limitations": "Some limitations have been addressed by the authors.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strenghts:\n- the paper is generally clear and well written\n- the method is well justified and appears to be mathematically sound\n- since it works at test-time, it does not require expensive retraining of state-of-the-art models\n- empirically, it clearly improves the baseline methods on standard benchmarks. Experiments present results on a large number of distortion and perceptual metrics.\n\nWeaknesses:\n- Reference [29] is a highly related method which seeks multiple perception-distortion tradeoff points with test-time iterations. Similarly to the authors work, it is a (progressive) mode-seeking strategy. The authors do not discuss similarities and differences of their work with respect to [29]. I think this point should be signficantly expanded. Moreover, since [29] essentially addresses the same problem, the authors should include experimental comparisons with it. At the current stage, it is unclear whether the authors' paper brings significant novelty or experimental improvements with respect to [29]. \n- An expanded discussion on computational complexity (and runtime experiments) would be useful to better understand the practical cost of the method", "was_revised": true, "final_justification": "The method seems sufficiently interesting and novel. The authors addressed my concerns in the rebuttal, so I lean towards acceptance.", "rebuttal": "Thank you for your valuable feedback and insightful comments on our work. Please see below for our point-by-point responses to your comments.\n\n### **Weakness 1**\n>Reference [29] is a highly related method which seeks multiple perception-distortion tradeoff points with test-time iterations. Similarly to the authors work, it is a (progressive) mode-seeking strategy. The authors do not discuss similarities and differences of their work with respect to [29]. I think this point should be significantly expanded. Moreover, since [29] essentially addresses the same problem, the authors should include experimental comparisons with it. At the current stage, it is unclear whether the authors' paper brings significant novelty or experimental improvements with respect to [29]. \n\nWe thank the reviewer for highlighting this related line of work. We now provide a detailed clarification of the conceptual and methodological distinctions between KDS and InDI [29], and will incorporate this discussion more prominently in the revised manuscript.\n\n1. **Different Motivations :** We would like to clarify the key difference in motivation: InDI is not a posterior mode-seeking algorithm. It performs a greedy, step-wise optimization on the _next latent state_, which does not guarantee convergence to  overall posterior. As a demonstration, Appendix B in the InDI paper [29] shows that the InDI estimate deviates from the MAP estimate for a simple Gaussian signal. In contrast, KDS directly seeks a consensus mode among the ensemble's predictions of the final clean output, a more direct objective for improving image fidelity.\n2. **Different Methods (Sampler vs. Training/Testing pipeline ):** KDS is a training-free, plug-and-play module designed for broad applicability, enhancing any pre-trained conditional diffusion model without retraining. This is distinct from InDI’s integrated training and sampling pipeline.\n3. **Direct Performance Comparison is Misleading:** The metrics reported in [29] are from a different dataset setting. InDI is expected to yield metrics similar to its baseline Conditional Diffusion Model, but with faster execution (as discussed in the InDI paper, Figure 12). Therefore, its metrics should closely align with our LDM-SR + DDIM baseline.\n\n---\n\n---\n\n### **Weakness 2 And Question 2**\n>An expanded discussion on computational complexity (and runtime experiments) would be useful to better understand the practical cost of the method\n>How complex is the proposed method in terms of runtime compared to the baseline without KDS? \n\nThank you for your question. We included a runtime comparison in Table 4 of our initial manuscript. \nAs detailed in Table 4, our KDS method with N=10 particles takes 28.4 seconds. This runtime is nearly identical to a standard 10-particle ensemble without KDS (which takes 27.3 seconds for a Best-of-N selection). This marginal (**~4%**) increase in runtime is a highly favorable tradeoff for the significant improvements KDS provides in both fidelity and perceptual quality.\nWe will make sure to clearly reference this study in the revised manuscript for better visibility.\n\n---\n\n---\n\n### **Question 1**\n>The experimental results seem to suggest that both distortion and perception metrics are improved by KDS. This is somewhat counterintuitive as it goes against the perception-distortion tradeoff principle, and does not happen in other works like [29] where improving one degrades the other. I wonder if the authors have an explanation for this? Could it be that the chosen baselines are rather weak? (e.g. [29] reports significantly better PSNR and FID numbers on DIV2K) \n\n\nThanks for the question. Please find our responses below.\n\n\n**Adherence to the P-D Tradeoff:** KDS does not violate the perception-distortion tradeoff principle but rather operates on a superior tradeoff curve. As Figure 6 in our submission demonstrates, adjusting the KDS bandwidth hyperparameter (h) allows for a trade-off between distortion and perception along the improved frontier. A larger 'h' value enhances distortion performance at the cost of perception, and vice versa.\n\n\n**Explanation for Simultaneous Improvement:** Standard diffusion samplers often introduce artifacts (e.g., unnatural textures, misplaced details). These artifacts degrade _both_ perceptual quality (increasing LPIPS) and fidelity to the ground truth (lowering PSNR). KDS leverages the \"collective wisdom\" of an ensemble to identify and suppress these spurious features. By producing a cleaner, more robust restoration that is closer to the true data manifold, it simultaneously reduces distortion and improves perceptual quality.\n\n\n**Strength of Baselines and Comparison to [29]:** The baselines we use (e.g., DiffBIR (ECCV 2024), SeeSR  (CVPR 2024)) are strong and SOTA for these tasks. The superior metrics reported in [29] are **not directly comparable** due to different training and testing protocols. As stated in our previous response, InDI [29]'s goal is to match its own baseline's quality with faster sampling. A fair comparison would therefore be between InDI's results and our _baseline_ LDM-SR results, over which KDS provides a significant and consistent improvement.\n\n---\n\n---\n\n### **Question 3**\n>It seems that 1x1 is the best patch size. Is this due to the curse of dimensionality for even moderately larger sizes or is it due to the low number of chosen particles? Would larger patches with a lot more particles work better (without caring for complexity)?\n\nWe appreciate this insightful question regarding the tradeoffs between patch size and ensemble size. Our current default choice of 1×1 patches with N=10 aims for a practical balance for most use cases, but we agree that richer patch interactions (with increased compute budget) are worth pursuing.\n\nAccording to your suggestion, we conducted a new ablation study, increasing the number of particles  (10 -> 40) for larger patches (1x1 -> 4x4), and observed an improvement in performance. Therefore, increasing the number of particles and using larger patch size can enhance performance by mitigating the curse of dimensionality.\n\nWe will include this observation and study in the revised appendix.\n\n---\n\n---\n\n**Table: New Ablation Study on Patch-size**\n\n| Method                             | PSNR   | SSIM   | LPIPS ↓ | FID ↓  |\n|------------------------------------|--------|--------|---------|--------|\n| DDIM                               | 22.05  | 0.531  | 0.307   | 20.89  |\n| + KDS (patch-size = 1, N = 10)     | 22.52  | 0.555  | 0.289   | 20.33  |\n| + KDS (patch-size = 4, N = 40)     | 22.70  | 0.563  | 0.281   | 20.08  |"}]

Author rebuttal:

Meta-review: This is a technically solid and well-presented paper that introduces a practical, training-free sampling method for diffusion-based restoration. The method is broadly applicable, improves multiple metrics consistently, and is conceptually intuitive.

Three reviewers (XDVv, S4WT, XT8v) were satisfied post-rebuttal and leaned towards acceptance, citing novelty, robustness, and practical impact. Reviewer XT8v found the work lacking in theoretical rigor, thus giving a borderline rating. The AC found that the rebuttal has sufficiently addressed this concern.

One reviewer (nw3F) leaned towards rejection, highlighting significant overlap with recent related works on mode-seeking and log-density control, questioning the use of KDE instead of the estimated score function and statistical significance of the results. The main contention lies in how distinct and impactful KDS is relative to methods pointed out by reviewer nw3F, in particular

[1] Karczewski et al. "Diffusion Models as Cartoonists: The Curious Case of High Density Regions" (ICLR 2025)
[2] Karczewski et al. "Devil is in the Details: Density Guidance for Detail-Aware Generation with Flow Models" (ICML 2025)

The authors addressed these concerns in the rebuttal with empirical evidence, which included a full hyperparameter sweep, confirming that KDS achieves a superior performance. Reviewer nw3F raised the score but still leaning to reject, stating that the paper requires a non-trivial revision when the new baselines need to be included in the discussions and empirical comparisons. 

The paper has been discussed with SAC. Both the AC and SAC were convinced that there are differences between the cited recent works and this submission. The comparisons against [1] and [2] showed that the proposed method is superior in performance. As suggested by reviewer nw3F, the authors must include the suggested experiments and provide a detailed discussion of these works in the revised manuscript.
Decision: Accept (poster)