CL-DPS: A Contrastive Learning Approach to Blind Nonlinear Inverse Problem Solving via Diffusion Posterior Sampling
Authors: Linfeng Ye
Shayan Mohajer Hamidi
Mert Pilanci
Konstantinos N. Plataniotis
Venue: ICLR 2026 Poster (2026)
Keywords: Diffusion Models
Blind Inverse Problems
Contrastive Learning
Primary area: applications to computer vision, audio, language, and other modalities

Abstract: Diffusion models (DMs) have recently become powerful priors for solving inverse problems. However, most work focuses on non-blind settings with known measurement operators, and existing DM-based blind solvers largely assume linear measurements, which limits practical applicability where operators are frequently nonlinear. We introduce CL-DPS, a contrastively trained likelihood for diffusion posterior sampling that requires no knowledge of the operator parameters at inference. To the best of our knowledge, CL-DPS is the first DM-based framework capable of solving blind nonlinear inverse problems. Our key idea is to train an auxiliary encoder offline, using a MoCo-style contrastive objective over randomized measurement operators, to learn a surrogate for the conditional likelihood \$p(\boldsymbol{y} | \boldsymbol{x}\_t)\$. During sampling, we inject the surrogate's gradient as a guidance term along the reverse diffusion trajectory, which enables posterior sampling without estimating or inverting the forward operator. We further employ overlapping patch-wise inference to preserve fine structure and a lightweight color-consistency head to stabilize color statistics. The guidance is sampler-agnostic and pairs well with modern solvers (e.g., DPM-Solver++ (2M)). Extensive experiments show that CL-DPS effectively handles challenging nonlinear cases, such as rotational and zoom deblurring, where prior DM-based methods fail, while remaining competitive on standard linear benchmarks. Code: \url{https://anonymous.4open.science/r/CL-DPS-4F5D}.

Published as a conference paper at ICLR 2026
CL-DPS: A CONTRASTIVE LEARNING APPROACH TO BLIND NONLINEAR INVERSE PROBLEM SOLVING VIA DIFFUSION POSTERIOR SAMPLING
Linfeng Ye Shayan Mohajer Hamidi"; Mert Pilanci", Konstantinos Plataniotis
'University of Toronto, 2Stanford University
'linfeng, yrffenngyeemmill. ye@mail. wtlll.utoroo utoronto.cco ca; kostas utoronto.ca
2 smohajer, pilanci (est edu
(a)
(b)
(c)
(d)
(e)
(f)
Figure Results of blind rotational deblurring, challenging nonlinear inverse problem: (a)
ground truth, (b) Ootation-blurres measurement, and restored images using (c) BlindDPS (Chung
al. 2023a), (d) FastEM (Laroche et al. 2024), GibbsDDRM (Murata et al. 2023). and (f)
CL-DPS (ours). All methods fail catastrophically except for CL-DPS.
ABSTRACT
Diffusion models (DMs) have recently become powerful priors for solving in- verse problems. However, most work focuses on non-blind settings with known measurement operators, and existing DM-hased blind solvers largely assume lin- ear measurements, which limits practical applicability where operators are fre- quently nonlinear. We introduce CL-DPS, contrastive learning framework for diffusion posterior sampling that requires no knowledge of the operator pa- rameters inference. To the best of our knowledge, CL-DPS is the first DM- based framework capable of solving blind nonlinear inverse problems. Our key idea is to train auxiliary encoder offline, using MoCo-style contrastive objective over randomized measurement operators, to learn a surrogate for the conditional likelihood p(y) During sampling, inject the surrogate's gra- dient as guidance term along the reverse diffusion trajectory, which enables posterior sampling without estimating or inverting the forward operator. We further employ overlapping patch-wise inference to preserve fine structure and a lightweight olls-consistency head capture color statisties. The guidance is cumpler-agnostie and pairs well with modern solvers (e.g., DPM-Solve+++ (2M)). Extensive experiments show that CL-DPS effectively handles challenging nonlinear cases, such rotational and zoom deblurring, where prior DM-based methods fail, while remaining competitive on standard linear benchmarks. Code: https::/ http://unnnnmmuuu..ummm 4oper. wwwenceeeeeeeceececec r/ CL- SBL--DS--- 4F5D
Authors contrihuted equally.

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Published as a conference paper at ICLR 2026
INTRODUCTION
Inverse problems pervasive across many fields, with applications in medical imaging (McCann ct al. 2017] Jin et 2017), computational photography (Tonolimi et al. 2020 Ongie et al. 2020), and seismic imaging (Hosseini & Plataniotis 2020; Zhdanov 2002). The goal is to recover the original signal Ey from a corrupted measurement y produced by forward operator Ag (.). Depending on whether this operator is available, inverse problems are categorized as (i) non-blind, where Av is known, or (iv) blind, where Av is unknown and must be estimated jointly with Vo. making the problem substantially more challenging.
Inverse problems are inherently ill-posed, often relying heavily on data priors p(æo) for accurate computation. Recently, diffusion models (DMs) have emerged as powerful tools for solving inverse problems due to their remarkable ability to capture complex data distributions p(æo) (Song et al. 2023 Chung et al. 2023b; Dou Song 2024). A straightforward approach to leveraging DMs for solving inverse problems involves training a conditional DM to directly estimate the posterior p( (a yv) via supervised learning. However, this method can be computationally intensive, as it requires training separate DMs for each distinct measurement operator Av.
To overcome this limitation, a practical alternative uses a pretrained, unconditional DM as a prior for p(æ0)) and combines it with a likelihood term inside diffusion posterior sampling (DPS) scheme. During sampling, the DM maintains a latent variable Dy that represents the current state along the reverse diffusion process at step t. Posterior guidance needs a surrogate for the intractable likelihood p(y (.), which recent work approximates in various ways (Chung et al. 2023b Song et al. 2023
Nevertheless, most DM-based inverse solvers remain limited to non-blind settings where the measure- ment operator A₂ is assumed known (Chung et al. 20236 Song et al.) 2023). However, accurately obtaining the measurement operator is often difficult or infeasible real -woldd applications (Chung) eet al. [023a] Laroche et al., 2024 Di et al.] 2024). Recently, efforts have emerged in the literature to address blind inverse problems using DMs. In particular, (Chung et al. (2023a) introduced BlindDPS, a method that trains a DM specifically for the blur operator. While this approach benefits from widely available pretrained DMs for signals such as images and audio, it requires additional training of a DM for the parameters of the relevant linear operators, significantly limiting its practical applicability. Murata et al. (2023) proposed GibbsDDRM, which constructs a joint distribution over the data, mea- surements, and linear operator, addressing the problem through posterior sampling using variant of the Gibbs sampler. Similarly, [Sanghvi SinilarlyySaanghvi]] (2024) estimated the kernel parameters to tackle deconvolution problems. However, these methods are fundamentally restricted to addressing only lincar inverse problems, as they assume that Ay is a convolution operator (Chihaoui et al., 2024) Sanghvil [2024] Murata al., 2023 Chung et al. 2023a). In practice, however, many blind inverse problems involve operators that are nonlinear, rendering these approaches inadequate for such cases.
To address the aforementioned limitation, propose CL-DPS, method based contrastive learning for solving blind inverse problems via diffusion posterior sampling. To the best of our knowledge, CL-DPS is first DM-based method that solve blind nonlinear inverse problems, without requiring knowledge or estimation of the operator parameters. Concretely. in CL-DDS, an auxiliary encoder first trained offline using a modified version of MoCo (He al.) 2020), contrastive learning (CL) technique. The role of this auxiliary encoder is to estimate the likelihood (az))))))))))))))))) without knowing the measurement Ag. Then, during inverse problem solving, we perform inference with this auxiliary encoder to estimate p(y/az;) which is then used to guide the reverse path of the diffusion process. To further improve the auxiliary encoder's accuracy in estimating p(y aa:). we introduce novel overlapping patch-wise inference method that divides the images into patches during the inference stage.
We evaluate CL-DPS on FFHQ (Karras et al. 2021), AFHQ (Choi et al. 2020), and ImageNet (Deng ctt al.) 2009) under blind linear and nonlinear mcasurements. In the nonlinear setting (c.g., rotation blur). prior DM-based methods fail, while CL-DPS restores high-quality images (see Figure I In the linear setting, CL-DPS is competitive with state of the art. Our main contributions are:
We introduce CL-DPS, diffusion posterior sampling framework that learns contrastive likelihood surrogate offline and plugs it in as guidance at test time. The auxiliary encoder is trained with MoCo-style framework over randomized measurements, so CL-DPS applies to both linear and
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nonlinear blind settings without knowing the measurement parameters at inference. In addition, lightweight color-consisten consistency head is deployed to capture color information.
We establish lemma (Lemmal I showing that, under energy -based formulation, the gradient of the contrastive log probability coincides with the desired likelihood gradient and becomes consistent the dictionary size grows. We further present denominator-awwe -wware variant, which yields results comparable to the simpler numerator-only update but incurs higher computational cost.
We introduce an overlapping patch-wiss inference strategy with an information -teeorrttt guarantee (Theorem M
stacking more overlapping patch features increases the mutual information between the signal and its encoded features, [(=; / (((55) (ef)))) }))-
Across blind linear and nonlinear tasks, including rotation blur, CL-DPS delivers strong quantitative and qualitative results. Ablations isolate the impact patch-wise inference, color regularization, and dictionary size, and confirm the efficiency of the numerator-only guidance.
2 RELATED WORK AND NOTATION
The use of DMs to solve inverse problems through posterior sampling has recently attracted considerable attention across various domains, including image denoising (Kawar et al. [2022] Hamidi & Yang, 2025b] a). compressed sensing (Bora et al.
Diffusion models for inverse problems:
2017
Kadkhodaie & Simoncelli 2021). magnetic resonance imaging (MRI) (Jalal et al.) (021).
scsre-based stochastic differential equations (SDEs) (Song et al.) (022), and variational methods
(Mardani et al.; Feng Bouman 2023). For non-blind inverse problems, methods such as diffusion posterior sampling (DPS) (Chung et al. 2023b) and pseudo-guided diffusion models (IIGDM) (Song)
ett al. 2023) leverage Tweedie's 's formula (Efron 2011) to approximate the smoothed likelihood.
Similarly, singular-valus decomposition based techniques (Kawar et al. (222I) are applied for related purposes. addition, [Hamidi et al. (2025) eliminates the need for constraint tuning or likelihood approximation by coupling data and measurement space diffusion posterior sampling.
On the other hand, for blind inverse problems, alongside the approaches discussed in Section T CChng
et al., 2023a Murata al. 2023 (aanghvi 2024), [Alkan et al. (2023) introduced Blind RED-Difi, an extension of the RED-DifT framework (Mardani et al.). This method employs variational inference to jointly estimate both the latent image and the unknown forward model parameters, addressing the challenges of unknown measurement operators. Recent work applies diffusion priors to amortized variational inference for inverse problems (Lee al. 2024), training a network end end to map y to posterior parameters under a specified forward model. Our setting is different: blind nonlinear operators (no operator estimates). We keep the diffusion prior frozen and use contrastive plug in likelihood surrogate to guide sampling from (I,, y).
We defer the discussion of contrastive learning to Appendix A
Notation: For a positive integer C, let [C] ف {L...., C}. Scalars are denoted by non-bold letters (e.g. and U), vectors by boldface lowercase letters (e.g. u). Denote by uzii] the i-th element of vector u. For two vectors u and v, denote by (u, v) their inner product. We use [C] to denote the cardinality of set C. (-) denotes the transpose operation. The mutual information between two random variables X and Y is denoted by I(X; Y).
3 BACKGROUND AND PRELIMINARIES
3.1 DIFFUSION MODELS
DMs generate data by reversing forward noising process. We adopt the variance preserving SDE (VP-SDE) (Song et al. 2020). which is equivalent to DDPM (Ho et 2020)
dæ
8. x dl
B BB.dw, dw,
with noise schedule 3, > O and standard Wiener process w. The data distribution is at t = 0 with Po Pdata and at 1. T the state is ærr NO. I).
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The reverse time SDE (Anderson 1982) is
dæ =
x V. log p(ax)) di + /Br
(2)
where flows backward and dw is reverse time Wiener noise. The score V logp(z;) is approxi- mated by neural network 89 trained with denoising score matching (Vincent 201
l" arg min (E.1) Bengr-Pan (e (2o) [1 [ () Var, log pog ao mo)ll
(3)
After training we use Sø- (xt; f) as estimate of the score in Equation 2 Discretizing Equation
yields the DDPM sampler. We use ar - 3, and a; - I 0y the discrete schedule.
3.2 DIFFUSION MODELS FOR SOLVING INVERSE PROBLEMS
We observe y E IR from an unknown ætp E Rd via
A +
(4)
where As is a linear or nonlinear measurement with unknown parameters &b and n is Gaussian with zero-mean and covariance I²I. The regime m < is ill-posed and requires prior on zo-
Using diffusion prior sample from the posterior by modifying the reverse SDE to include the likelihood term
da: = a - Be (v. loge( + Va, logp(y ax))] di + v B dw
(5)
which follows from
V m, log p( | Vz, logp(æ;) + V log xt).
(6)
The prior score Va, log ((x)) is given by the pretrained network sg-. The bottleneck is the likelihood term Vn, log p(y xx)) which time dependent and intractable when vp is unknown.
Prior work often assumes known operator for likelihood evaluation (Chung et al. 2022a 2023b) We address the blind case by learning contrastive likelihood surrogate offline and using it guidance inside the sampler, as detailed in Section
3.3 MOMENTUM CONTRAST LEARNING
Contrastive learning learns representations by pulling together positives and pushing apart negatives (Hadsell et al., 2006 Tian et al.) 2020). MoCo (He et al. 2020) implements this as dictionary lookup with two components: a large queue of keys that serves as dynamic dictionary and momentum encoder that produces stable keys. Let fy be the query encoder with parameters O,, and fú the key encoder with parameters 0₂ updated as 0₃ < mQ, + (1 - m))g. where m E [0, 1) is the momentum coefficient. Given query q = f-) (-) a positive key h4 fx(-) and K negative keys {k,) from
the queue MoCo minimizes the InfoNCE loss (Oord et al. 2018)
expl (g. k )/T)
Cy - llog
Σ exp (ep((g) k.)/T)
with temperature 7 > 0. In method we set q f(x,) the positive key to k4 f(Hsy) and use the MoCo queue to approximate the dictionary of negatives.
METHODOLOGY
As discussed in Section 3 diffusion posterior sampling relies on the likelihood term p(y (x)) and its gradient with respect to We approximate this term with an auxiliary encoder f. trained offline to provide a surrogate across a range of measurement operators since the parameters zb are unknown at inference. During reverse diffusion, f supplies the likelihood gradient guide the sampling trajectory. Section detaiis the contrastive learning procedure used to train f. Section 4.2 shows how it is applied at inference, and Section 4.3 presents the full CL-DDPS algorithm.
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4.1 TRAINING THE AUXILIARY ENCODER
4.1.1
CONTRASTIVE LEARNING AS LIKELIHOOD ESTIMATION
Using Bayes' formula, the likelihood p(y (z:) can be written as
p(y (xt) p at) p(y, vza)
p(a))
J p(Ú m.)) dÿ
To compute Equation 8 first obtain numerical representation of its numerator, p(y, 2t). Specifically, following (Oord et al. 2018: Li et.
202T). we approximate it with an energy score expl (Zz)) f(v)) / 7). where > 0 is temperature.
learned by a neural encoder as: pyy. 2t)
The denominator in Equation 8
p(ū 2 ddy. is generally intractable. Thus, we rely on an follows: f [' at )d v Eÿer p(ù, æx), where Y is sufficiently large set. This allows us to numerically approximate p(y xx)) as:
approximation method, using a finite
f() Eger (f,(((())) (())))
plu xx)
(9)
This suggests training f by maximizing the log of Equation Equivalently we minimize the negative log likelihood surrogate
Cplyie)) = log exp(([(x.), (y))/T)
(10)
Ejer exp((((a) (x.), (())//T)
Comparing Equation (10) with the InfoNCE loss in Equation shows that the standard contrastive objective direct estimator of this surrogate when the query is q f(x,) and the keys are (kieej C Y. With a large dictionary as in MoCo, the queue of size K provides a practical approximation the population Y.
To justify that this contrastive surrogate serves as the likelihood term required by diffusion posterior sampling, we state the following Lemma that links the contrastive log probability to the conditional likelihood gradient. The full version with assumptions and the proof are provided in Appendix B
Lemma (Contrastive likelihood gradient, short version). Let t > 0 and define
sc y) & (f(z)), f(y)) /T.
(11)
For a finite dictionary Y with y E Y define the softmax surrogate exp(sr(x, y)} Pry ( Eger exp(si(x; 9))
(12)
Then,
Va, log (y xx) Va, v) Epe y (U xe) st(, (), BEY
(13)
Moreover, under the energy model p(y æx) exp(sr(xx b)} with mild integrability conditions stated in Appendix B and with Y, drawn i..dd. and augmented so that y E Yn
Va, log Pr.y., (y xz)
Va, log.ly xt).
(14)
4.1.2 TRAINING THE AUXILIARY ENCODER FOR DPS
To optimize Equation 10p we ideally need pairs (æ,, y) drawn from the true measurement process since inference will condition the observed y. During training the measurement parameters &b of the operator family Ag are unknown and vary at test time, so we replace y by surrogate synthetic measurement Ysy generated by sampling & from prior Py. This trains the encoder approximate the likelihood across range of operator parameter settings. To this end, given clean image Ty we synthesize the training pair (I,, Msya) at randomly chosen time
Ysya A
D3 vāa Do + /I 04
(15)
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where 2b Py and (N,, The diffusion schedule satisfies ā = =I;;- a, with 0 aj 1. Examples of Ab include Gaussian, motion, rotation, and zoom blur, where the operator parameters & are sampled from the prior Pv. which commonly chosen as a uniform distribution over a specified range of blur intensities. At each training iteration, construct distinct pair (æt, Vyn). Note that although f does not take explicitly, its input æ; is drawn at timestep t, so the surrogate pyy xxa) (and its gradient) are implicitly t-dependent.
The contrastive dictionary
formed by the current mini-batch's embeddings (EBDs) together with queue of EBDs from previous batches; the EBDs are produced by momentun-uuddated encoder, as in MoCo. We train the encoder with an InfoNCE-style objective (p(ulllae))
In practice, the contrastive objec- tive can become insensitive to color information, often resulting in hue or brightness shifts in reconstruc- tions. To mitigate this issue, we add lightweight color-consistency head (CCH), denoted by H.s on top of the auxiliary encoder. The CCH trained to predict the global color statistics of the input. Formally, let æ, E RCx NxxN2xx N2 and define its spa-
Momentum Encoder
Memory Queue
Projection
y sym
ema
ema
sg
Projection & H₂(+)
EBD
pp( (m;)
Encoder
A
H.A.)
Loe
AP(e
EBD
HL(ev)
t
vI A:1
Sigmoid
Projection
tial average
Conv 1X1
Linear
Hc(-)
1 N1
Relu
[AP(z:)].
Tickj
Conv 1X1
Trainable Components
The color-consistency head Hz(x):) E RC is implemented as two-layer convolutional module with global pooling followed by a sigmoid activation. We define the color-consistency loss as
Overview of the training process for the auxiliary encoder. The figure also illustrates the structure of the linear projection head and the color-consistency head (CCH). The CCH is a two-layer convolutional network that encourages the model preserve the color information of the input during training. & indicates trainable components, sg stands for stop gradient, and ema for exponential moving average.
Figure 2:
(oc(#x) (a.)) AP(x:))l2
To train the auxiliary encoder f to
approximate the likelihood and cap-
ture color information, we optimize the following combined loss:
LCL.DPS Ep( Lco(w)))
(16)
where A >> 0 balances likelihood estimation and color preservation. The loss in Equation (16) is averaged over the mini-batch. The CCH is used only during training and discarded inference. At test time, Ylyn is replaced by the observed measurement y, and the pretrained encoder f provides surrogate likelihood guidance within CL-DPS. The framework for CL-DPS depicted in Figure 2
Remark 1. CL-DPS adds small, one time auxiliary training stage enable the blind nonlinear setting. This stage is lightweight, and the encoder is trained once for a given forward setting and then reused for all measurements and noise realizations. The procedure remains zero shot for the downstream task because it uses only synthetic pairs generated from the diffisson prior and the measurement simulator, no extra labels or human annotations are needed. In practice this offline cost is far smaller than pretraining or fine tuning a DM or training a supervised reconstructor, and per image inference remains on the same order standard DPS (see Appendix K}. The added training enables, to our knowledge, the first DM-bused solution to blind nonlinear inverse problems without estimating operator parameters, which prior training free methods do not handle.
4.2 LIKELIHOOD ESTIMATION USING AUXILIARY ENCODER
After training with Equation (16). the encoder provides surrogate for p(y));)) that we use inside diffusion posterior sampling. However, the convolutional encoders often compress low level details
(Yang al. 2025 Chi et al. 2025 Tishby & Zaslavsky 2015 Ye et al. 2024 Yang & Ye 2024).

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Published as a conference paper at ICLR 2026
which reduce the granularity needed for inverse problems. We therefore overlapping patch-wise inference scheme that increases how much information the encoder output retains about its imput.
4.2.1 OVERLAPPING PATCH- WISE INFERENCE
Given image E RR we partition it into Lg overlapping x n patches with stride < N + +1 Then, we run f on each patch and Ni -i +
denoted by {p" ]E]] where La stack the features
((P]bss((.))
(ST(P)) (ptt))
(17)
To quantify retained information we use mutual information [ar; (a)). The next result shows that stacking more overlapping patch features increases this quantity.
Theorem 1. Let æ E RNNNN Na be any random image. Fix a patch size n X n and a stride < n. Let f: TR - TRd act pitch-wise. For integers U V extract U and V overlapping patches and form stacked features as above. Then,
1(x; [(P be(u)) < I(2; ((pf)bav))
(18)
The proof deferred to Appendix C In words, denser overlapping patching makes the encoder output more informative about æ.
Corollary 1. Adopt the notation of Theorem ((PFT) E Rdd Then for every integer U 2 1, we have 1 f(x)) S x;; x ((p] heww))
For single extracted patch (U = 1), write fx) &
We further complement Theorem and Corollary with variance- reduction analysis for the guidance estimator, simple design rules, and an empirical study in Appendix D
Algorithm CL-DPS (ours)
Input number of steps I', measurement y. noise schedule (ā.). pretrained encoder f-). ).. step size r) > 0, number of overlapping patches U.
2: Pr ~ N(, I).
// initialize with Gaussian noise 3: Extract U overlapping patches from once.
During likelihood estimation we ap- ply the same patchification to the measurement y and use the stacked features to score consistency.
cache measurement features
(p") JE[! f- y.
5: for l = T .... do
6: å + s0 (xi, t).
// score model estimate of Va, log p(z))
ao Vou xt + (1 ao)a). Tweedie posterior
Remark 2. Patch-wise ideas exist in prior work. Hu et al.) (2024);
8:
z NO I
Zhang et al. (2025) train a patch- based diffusion prior and aggregate patch scores with positional encod-
9:
x
xc +
Vas-18e 88
+ 0.z. DDPM update
1-6ee
1-6e
10: Extract overlapping patches from we.
ing to form whole-image prior
11:
(p) be(u +- Et.
score. WWang al. propose DDNM, a zero-shot restoration method that uses off-the-shelf diffusion prior and enforces data consistency by modifying only the null-space com- ponent for linear measurements, without auxiliary training. In contrast, our overlapping patch-wise step is neither a prior nor null-space projector. It extracts stacked local features that are fed to auxiliary encoder learning contrastive likelihood surrogate for DPS guidance.
12: x: x - Vas (s((p)) bew)).
I/ contrastive guidance
13: end for
14: Output
4.3 ALGORITHM FOR CL-DPS
After training with Equation [6). we keep only the encoder f and use as likelihood surrogate inside diffusion posterior sampling. We integrate f into DPS (Chung et al. 2023b) with overlapping patch-wise features from Section 4.2 The only change relative to unconditional sampling a contrastive guidance step that adds an estimate of V log p(y x;) at each time t. The procedure is summarized in Algorithm|] [ In line 12 of Algorithm [ we adopt an mnergy-guidance view: treat the contrastive unnormalized likelihood and use only the numerator gradient, following prior
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Table Nonlinear blind inverse problems: Blind rotation and zoom deblurring results on the FFHQ. AFHQ and ImageNet datasets. Only CL-DPS achieves high-quality image restoration; other methods fail. Bold and underlined values denote the best and second-best results, respectively.
Rolation
FFHQ (256
256)
AFHQ (256 256)
ImageNet
Method
PSNR
FID
LPIPS
PSNR
FID 1
LPIPS
PSNR 1
FID
LPIPS
CL-DPS
(SPE)
22.74
33.66
0.302
21.46
36.96
0.319
20.05
45.10
0.340
CL-DPS (UNT)
22.27
36.55
0.315
21.61
39.81
0.330
19.92
49.23
0.352
FasHFM wwr s00
15.96
268.4
0.597
11.57
289.2
0.684
13.90
337.8
0.721
BlindDPS eve Brotu
16.87
343.8
0.552
13.25
200.5
0.674
11.25
392.4
0.895
Gibb:DDRM po
18.43
236.6
0,565
15.24
263.5
0.628
12.24
311.2
0.781
Pan-fo mRM
14.63
327.7
0.629
13.41
227.8
0.895
11.52
340.7
0.862
Zoom
FFHQ (256
256)
APHQ (256
258)
ImageNer
Method
PSNR
FID L
LPIPS
PSNR
FID |
LPIPS
PSNR f
FID
LPIPS
CL-DPS (SPE)
20.68
42.61
0.435
19.63
57.54
0.468
18.56
55.30
0.481
CL-DPS (UNI)
20.31
46.83
0.448.
19.23
61.06
0.480
18.07
59.53
0.492
FastEM
17.68
303.4
0.623
15.69
310.1
0.797
12.76
331.4
0.754
BlindDPS evi
16.39
292.9
0.784
11.75
279.6
0.607
11.96
348.2
0.812
Gibb:DDRM W0 300
15.45
327.4
0.802
14.57
280.5
0.549
11.14
299.8
0.702
Pan-fy 1960
11.52
392.2
0.715
12.41
292.8
0.851
9.452
347.2
0.743
A denominator alternative is provided in Appendix E where we show that yields essentially the same reconstruction quality but with higher computational cost.
unnormalized-energe guidance (Lu et al.) 2023 Du et al. 2023).
We note that CL-DPS can be seamlessly combined with alternative diffusiin sampling processes. For example, in Appendix we pair CL-DPS with DPM-Solver+4 (2M) (La et al. (025) providing the algorithm, implementation details, and representative results.
Remark 3.
EXPERIMENTS
5.1 EXPERIMENTAL SETUP
Implementation details. Implementation details for CL-DPS, including training the auxiliary
encoder, and implementation details for baseline methods provided in Appendix G
Datasets. We use Flickr-faces-HO (FFHQ) 256 x 256 dataset (Karras et al. 2021 and animal
faces-HQ (AFHQ) 256 x 256 dataset (Choi et al. 2020) and ImageNet (Deng al. 2009).
Pretrained diffusion models. We leverage pretrained score functions from Chung et al
2022b
Evaluation metrics. We Fréchet inception distance (FID) (Heusel al. 2017) learned
perceptual image patch similarity (LPIPS) (Zhang et al.) 2018) and peak signal-to-noise ratio (PSNR) between the original image and the reconstructed image the evaluation metrics.
Choice of benchmarks. For the blind nonlinear setting, to the best of our knowledge there are no prior DM-based methods; we therefore compare against the strongest blind linear DM-based solvers (FastEM (Laroche et al. 2024), BlindDPS (Chung et al. (023a)). GibbsDDRM (Murata et al. 2023) and include one classical non-DM baseline, Pan-lo (Pan al. 2017). ). For the blind linear setting,
evaluate against the full set of seven baselines: SelfDeblur (Ren et al. 2020). DeblurGANv2 (Kupyn
al. 2019). Pan-Ly (Pan et al. 2017). BlindDPS (Chung et al. (023a). FastEM (Laroche al.
2024). LatentDEM (Bai et al. (025). ) and GibbsDDRM (Murata al.) (203)). The last four methods use DMs. CL-DDSS not the only method that uses synthetic degradations for training. In particular, BlindDPS and FastEM train operator priors synthetic blur kernels, while DeblurGANy2 is trained on synthetic degraded-clean image pairs.
For CL-DPS, we evaluate two training regimes: (?) Universal (CL-DPS (UNI)) ), using single encoder trained jointly across all operator families; and (ii) Specialist (CL-DPS (SPE)), training separate encoder offline for each family A& (e.g.. distinct encoders for rotation and zoom blur). At inference, we select the encoder for the detected family while still treating z as unknown. See Appendix H for justification.
8

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Published as a conference paper at ICLR 2026
Table 2: Linear blind inverse problems: blind motion and Gaussian deblurring results. Bold and underlined values denote the best and second-best results, respectively.
Motion
FFIQ (256 x 256)
AFIQ (256
256)
ImageNet
Method
PSNR
FID
LPIPS
PSNR1
FID
LPIPS,
PSNR
FID1
LPIPS_
CL-DPS (SPE)
26.33
27.44
0.117
24.06
26.25
0.186
22.27
40.35
0.131
CL-DPS (UNT)
26.17
28.90
0.124
23.72
0.191
21.88
43.20
0.135
SelfDehlur evi
10.83
270.0
0.717
9.082
300.5
0.768
9.542
320.1
0.775
DeblurGANy2 Ice
17.75
220.7
0.571
17.64
186.2
0.597
18.40
260.2
0.561
Pan-fo maad S0r3
15.53
242.6
0.542
15.34
235.3
0.627
14.92
275.2
0,585
BlindDPS meta
22.24
0..881
20.92
19.59
0.341
FastEM wack
24.68
34.52
0.340
21.60
50.80
0.315
18.03
38.24
0.345
LalenDDEM
22.65
37.10
0.167
20.32
45,61
0.285
18.55
42.53
0.295
GDM KMM 1001
25.80
38.71
0.115
22.01
48.48
0.197
17.10
43.22
0.240
Gaussian
FFHQ 256)
AFHQ (256 256)
ImageNer
Method
PSNRT
FIDD
LPIPSL
PSNR1
FIDL
LPIPS
PSNR1
FIDL
LPIPSL
CL-DPS (SPE)
26.42
26.65
0.218
21.76
20.16
0.225
22.05
34.11
0.255
CL-DPS (UNI)
26.35
27.05
0.228
24.40
22.25
0.237
21.85
36.90
0.268
SelfDeblur ovm
0000
11.36
235.4
0.686
11.53
172.2
0.662
10.22
280.5
0,740
DeblurGANv2
3018
19.69
185.5
0.529
20.29
86.87
0.523
21.56
60.31
0.393
Pan-fo moaa
19.94
92.70
0.415
21.41
62.76
0.395
18.52
0.462
BlindDPS
24.77
27.36
0.233
23.63
20.54
0.287
19.59
51.25
0.341
FastEM warN
23.15
30.25
0.375
22.95
32.15
0.295
17.51
36.01
0.285
22.75
30.53
0.365
21.57
38.24
0.296
19.31
38.25
0,273
GibbsDDRM
26.34
34.12
0.426
23.12
42.75
0.314
19.63
38.10
0.355
5.2 RESULTS
Nonlinear deblurring. We consider rotation blur and zoom deblurring tasks as nonlinear inverse problems. For rotation blur, randomly choose a rotation center per image, set the rotation angle in [10", 3,30), and apply a random weight to the rotation trajectory. For zoom blur, we set the center of the image as the focal point of the zoom, then apply zoom factor in [1,3].
The qualitative results for the rotational deblurring task using benchmark methods and CL-DPS are shown in Figure 0 As observed, CL-DPS is the only method capable of successfully recovering the ground truth images, while all benchmark methods fail to do so. Qualitative results for the zoom deblurring task are provided in Appendix N Also, the quantitative results presented in Table [ The results on all three datasets show the significant superiority of CL-DPS over benchmark methods. Using Univers encoder for all measurements leads slight performance drop compared to the family-operator setting, yet CL DL-DPS (UNI) still remains far ahead of all baselines.
All DM-based benchmark methods fail in nonlinear settings because they assume that An is a convolutional operator an assumption that cannot be remedied with a simple modification.
Linear deblurring. For linear deblurring, consider Gaussian and motion deblurring. Specifically, following (Bai et al. 2025; Laroche et al. [2024] Murata et al.) 2023), we apply the Gaussian blur kernel with the size of 61 61 and standard deviation of 3.0. Also, the motion blur kernel is generated randomly using an open-soure code with kernel size of 61 x 61 and intensity of 0.5. These kernels are convolved with the ground truth image to produce the measurement.
Table 2 summarizes the quantitative results for Gaussian and motion deblurring tasks. Compared to state-of oh-t methods, CL-DPS achieves competitive performance across various metrics under blind linear inverse settings. Notably, CL-DPS outperforms all the other methods in terms of PSNR and FID score on the FTHQ and AFHQ datasets when subjected to Gaussian blur. Further Qualitative results are provided in Appendix O
Comprehensive ablations on contrastive hyperparameters, overlapping patch- wise inference, and the color-consistency head are provided in Appendix M Wall-clock run times reported in Appendix

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Published as a conference paper at ICLR 2026
Oitiioary size
-64
20
|Y -256
Dictionary
G 6.95
[
10
|Y |||24
10
50
10
50
Figure 3: Toy gradient check on subsampled grid of timesteps. Left, angle between Jtne and Oour Right, norm ratio 1 lerellggellll l22 Larger |Y] yields better alignment. Details are in Appendix
6 TOY LIKELIHOOD GRADIENT CHECK
We validate that the surrogate guidance aligns with the true likelihood gradient in controlled synthetic setting (full setup and hyperparameters are in Appendix [[]. For a fixed diffusion schedule, subsample Ting 50 timesteps that equally spaced in log os At each we form toy observation y, with a linear operator Hい and Gaussian noise, then compare the closed-form gradient
Jwve (t)
Ve, log p(ye xx) or 2 (3: Hw.w.)
(19)
the gradient of contrastive softmax surrogate
Dsur (1)
Va,, (5(1 De())) S(w)) ((p))))))) )E(()))
(20)
We report the angle 02 = cos
(Amme(t). Que [[)mm (1)
and the norm ratio Pt
4ma (d)l
(P)
Figure shows that 0g decreases and approaches one as i. increases. Larger dictionary sizes further improve alignment.
Y
7 CONCLUSION AND FUTURE WORK
We proposed CL-DPS, diffusion-based method for blind inverse problems with unknown mea- surement parameters. By training auxiliary encoder via a modified MoCo framework, CL-DPS estimates p(y(æ;) without access to the measurement operator and guides the reverse diffusion process accordingly. We further improved estimation accuracy by using color consistency head and deploying overlapping patch-wise inference. Experiments show that CL-DPS handles both linear and complex nonlinear settings, including tasks like rotational deblurring, where prior methods fail. One potential direction for future work is to design more efficient auxiliary encoder to further reduce inference cost.
LLM USAGE STATEMENT
LLM used only for grammar and wording edits; no generation of ideas, methods, analyses, results, or citations. Authors reviewed all edits and accept full responsibility.
REPRODUCIBILITY STATEMENT
We have taken steps to ensure our results are reproducible. All model and algorithmic details, training procedures, hyperparameters, evaluation protocols, and metrics specified in the main text. The appendix provides complete proofs, implementation notes, ablations, and additional qualitative results. An anonymized GitHub repository contains the source code and configuration files, and pre-trained checkpoints. All datasets used in our experiments are publicly available.
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A RELATED WORK
A. CONTRASTIVE LEARNING
As a versatile semi-supervised learning framework, contrastive learning learns a useful feature representation by clustering positive samples and dispersing negative samples. It achieves great success since instance discrimination has been proposed in (Wu al. 2018). Since then (Chen et al.. 2020alb) advanced the field by leveraging diverse data augmentation methods and using projection head during the contrastive learning process. (He et al., 2020) used momentum update mechanism to maintain negative sample queue to reduce the memory consumption. Later (Wang et al. 2021) further proposed dense contrastive loss for dense downstream task, specifically, (Wang et al.) 2021) can consistently outperform its bascline methods like (He et al. (2020), when transferring to downstream dense prediction tasks including object detection. For interested readers sceking further information, please refer to the survey paper (Gui et al. 2024).
B FULL VERSION OF LEMMA 1 AND ITS PROOF
Lemmal 1 Full Version.
Let be differentiable æ, and let t > 0. Define
st (x y) & (f(xi), / T.
(21)
For a finite dictionary Y with y E Y define
exp(sr(01 y)}
by xx)
(22)
Eger exp(s: [x 99)
Then,
Ve, log Pr xe)) = Va, sc(eee y) - Σ
T (ū x+) Va, 31(20, 1))
(23)
BEY
where
cxp(s:(24 97)}
Tr,y (ū ax) Exer exp(s1(2)) =)}
(24)
Assume the energy model
p xx) exp(s<(< |y}} / Z, (x,),
(25)
with base measure Y and partition function
Zz(x,) = / exppss(Iza 22} du(z).
(26)
Let Yn be i..dd samples from μ augmented that y (x)e))())))))) | St (2:l)-)) are y integrable then for fixed y and w;
Yn- If exp(s)(x.l)) and
Va, log Pa,ya (y xx)
a.s. Va, logp(y xz).
(27)
Proof. Define the empirical partition function
(,,((x)))
Σ
exp(s((x1) x)}
(28)
zEY
Then,
log pr,y (y) xx) sx ()) log (xx))
(29)
Differentiating with respect to gives
V x, Po Pr, xz) Vt (z) Ze.((x,)
Vol (z:))
(30)
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Published as a conference paper at ICLR 2026
The gradient of I is
Væ, Z1, = Σ
exp(st(2x Ø)} Ve, s1(2; g)
(31)
DEY
Therefore,
Va, Ze, = Σ
F xx)) Sc(t D)
(32)
Ze, (x:)
DEY
which proves the gradient identity.
For consistency consider the partition function
Ze(x,) = f /expls((0 x)} du(z).
(33)
Let g(2) Se 2). By the strong law of large numbers and the integrability assumptions
Σ
exp(s((xx
x)} a / exp(s x)} du(z))
(34)
Y
zEY,
exp(s1(xx [z)) ae, / /exp(s((er 1z)]((z) g(z) du(z),
Σ
(35)
YY
zeY,
Hence
TL.Y, (U xz) g(ū) a. fexp(s((2> 2)} g(2))) du((z)
Σ NEY
(36)
se x)} du(z))
The right hand side equals Eg Ez-pl (x.) [g(2]] Finally
Væ, poy æ.) Va, St y) Ez Ezmp( (m)) Vi [2)]
(37)
Combining the empirical identity with this limit shows
Væ, log Pr.y. xx) a.5., Va. logply xz),
(38)
which completes the proof.
C PROOF OF THEOREM
Proof. Define the additional patch collection
G(x) A (p so that f(pf))etv)) (Ipf ffe)))
Using the definition of mutual information and the chain rule for entropy,
I( x:: ((pfhevv)) = H(x) - | ((pfhaav)
H(x) - x bel))
where H(-)) denotes the entropy function. Since conditioning cannot increase conditional entropy,
(< f(ppT)ee)) < H ((x (epf((t))))))
Substituting gives
((p))) 2 H(x)- H (() = ((p;; (et)))
The argument uses only that the patch extractor and are deterministic functions of and that the entropies are well defined. No independence or distributional assumptions on æ are required. This proves the claim.
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Published as a conference paper at ICLR 2026
D FROM MUTUAL-INFORMATION MONOTONICITY TO PRACTICAL PATCH DESIGN
Theorem I concerns the clean image x and the stacked patch-wiss features f f(p33)) During sampling we operate on diffusion states and form a surrogate likelihood guidance using patch-wiss features. In this appendix we relate the monotonicity guarantee to the stability of that guidance and derive practical patch-selection rules.
D.1 VARIANCE CONTRACTION OF SURROGATE GUIDANCE
At reverse step f, let the per-patch contribution to the surrogate likelihood gradient be 6,(2)) E R. computed from the j-th overlapping patch feature. Define the averaged guidance
U
gu (a.))
Σ (x)))
(39)
U
Assume the following mild regularity conditions: (i) bounded second moments, Var(Ø;) 3 o² for all j. and (ii) weak dependence across overlapping patches, Ex2ıllCor(()) 25+x) < 00.
Proposition 1 (Variance contraction). Under the conditions above,
Var (gu(x:)) 3 T I
(40)
for a constant that depends on a² and the dependence sum. Consequently, the signal-to-ne to-isse ratio of gu increases with U, and the expected cosine similarity between gu and its mean increases toward one at a rate that saturates on the order of 1/U.
Proof. Write o; = (xx,) for brevity and let 3, = Φ - E, For any unit vector u E S
define the scalar sequence E = u Φú.- Then
U
U-1U-k
(
Var(u gu) = Var
M E3 j=1
= U
Var(E)) + 3 Σ Eor(E),()tt))
(41)
k=
Bound the diagonal terms using Var(E,) = u. T Var (f) u S 0². For the off diagonal terms, note that
(Cov(E)))+**] =Iu² Cov Cor (5+k) ml (C $ Pk-
(42)
Therefore
U-1U-k
U-1
Var(u gu) x T
U + 2 Σ Σ Pk) = U2
Uo² + 2 Σ
(U - k)pp))
(43)
k= j=1
1=1
U-1
8
02
Σ
s
02 + 2
Σ pk
(44)
Since the bound holds for every unit vector u, by the variational characterization of the operator norm obtain the matrix inequality in equation 40 that is
Var (gu
1 + 1
(02
U-1
Σ k=1
1
)pL) I
k
1 (
+ 2 pk k=1
8
I.
(45)
This completes the proof.
Corollary (Design implication). Increasing the number of overlapping patches U yields diminishing-rette returns improvements guidance stability. This explains why small to moderate overlaps deliver most of the gain while very dense overlaps plateau.
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D.2 A COMPUTE-AWARS PATCH POLICY
Let be the patch size and the stride. We use Theorem | as safety guarantee and choose granularity by the following rule. (i) Choose n so that salient structures span multiple patches, (ii) choose in the range n/4 to n/2 so that each pixel participates in several overlapping contexts, (iii) increase U until a held-out validation metric (for example PSNR or FID) saturates, then stop. This policy turns the monotonicity guarantee into practical selection strategy without over-allocating compute to excessive overlap.
E DENOMINATOR TERM: THEORY, DIAGNOSTICS, AND IMPLEMENTATION
In the main sampler we apply the contrastive guidance using a numerator-only update at line 12 of Algorithm]] | This section justifies that choice, reports an empirical magnitude diagnostic of the softmax denominator term, and provides the full denominator-aware variant for completeness. Let the logits be 5; = (f()),, (z) with probabilities
exp(s,)
= Σx exp(s*)
(46)
and let the softmax surrogate be
(()())
D.y xx)) = Egey exp (())/TT)
(47)
From Lemma j
Væ. log log (2y(t) xx)) = V
84
Σ
F
V
89.
(48)
of
With unit normalized features and comparable Jacobian scales one expects
Adenlle l
(49)
|num 2|
so a well trained encoder yields small ratio when n. is large.
Magnitude diagnostic on toy dataset. To verify the claim, reuse the toy setup and the subsampled diagnostic grid from Appendix For N = 120 held out toy images and Taing 50 timesteps, compute
= |1
(50)
9oum 2
at each t. Unless stated otherwise we fix |Y| = 256, temperature T = 0.07. &g feature normalization. 0.996. The left panel of all x Tdiag ratios into a single histogram. The right panel shows the per median with the interquartile range. Most mass lies between 0.08 and 0.20, and the median decreases mildly with t. This indicates that the denominator term typically much smaller than the numerator term in the guidance regime of interest.
MoCo queue length K 65536, and momentum
For completeness, the denominator-aware update replaces line 12 of Algorithm 1 with
Øt-1 2'_1
17
V de
s+
Σ
TLY (ū
sa(x の)
(51)
DEY
which requires evaluating fxx) against all ý E Y and taking a weighted sum of their gradients. This increases compute and memory per step.
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0.225
median
19cen// //1gumm
Igee Ptenl/lgwww pumml
IQR
0.375
0.150
0.125
0.300
0.075
0.00
0.05
0.30
0.15
0.30
0:35
50
ratio
Figure 4: Magnitude of the softmax denominator term on the toy dataset from Appendix [ Left, histogram of ri [[geenll2 //mmm/lmmlll aggregated over images and timesteps. Right, per median with interquartile range. Small ratios support the efficient numerator-only guidance used in the main sampler.
Numerical Comparison To quantify the effect of including the softmax denominator term in the guidance update, compare the default numerator-oni update with the full denominator-awei variant described above. Both methods are run with identical hyperparameters, timesteps, and random seeds ensure fair comparison. Results are averaged the same held-out test split and noise schedule used for Table
As shown in Table incorporating the denominator term leads to slightly higher PSNR and noticeably lower FID/LPIPS, confirming that the correction provides a modest but consistent benefit. However, this comes the cost of additional compute and memory per step since it requires evaluating the encoder against all E Y and forming weighted gradient sum.
Table 3: Comparison numerator-only and denominator-awee
aware aware contrastive guidance (Appendix E). Results are averaged over the same test split and noise schedule in Table 1. Denominator-awes guidance slightly improves perceptual quality (lower FID/LPIPS) but at higher compute cost.
FFHQ (256 x 256)
AFHQ (256 x 256)
ImageNet
Task
Variant
PSNRT
FID]
LPIPS
PSNRT
FID|
LPIPS]
PSNR
FID
LPIPS
Numerator-only
22.74
33.66
0.302
21.46
36.96
0.319
20.45
45.10
0.340
Rotation
Denominator-awen aware
22.79
32.95
0.297
21.52
36.99
0.315
20.52
44.35
0.336
Numerator-only 20.68
42.61
0.435
19.63
57.54
0.468
18.56
55.30
0.481
Zoom
Denominator-awan
20.73
41.72
0.429
19.70
56.40
0.471
18.60
54.35
0.474
Given the small empirical ratio r., the nearly identical quality, and the higher cost of the denominator- aware step, we adopt the numerator-only update in the main algorithm. If one observes large early step ratios 2 0.4, two practical mitigations are to lower the temperature for the first few steps to increase Y moderately, both of which increase _. and reduce the denominator scale.
F DPM-SOLVER++ FOR CL-DPS (SAMPLER DETAILS)
Our method augments each reverse step with a contrastive likelihood surrogate. This guidance is sampler-agnostic: it only requires evaluating a gradient Vz, Le,,(a y) at the current iterate. Replacing the first-order DDPM/DDIM stepper with a high-order solver reduces discretization error with and improves stability under strong guidance. We adopt DPM-Solver++ (2M) (Lu et al. with
Karras o-schedule.
Let æ; Voi No + o Our denoiser predicts noise (0 o.). The CL.DDPS guidance is computed in image space as g: Væ. {f (2))chees(z)) (patches(y))). where features of the measurement are cached once. To apply guidance consistently with an e-parameterized solver, we use the local relation 0z;/0e de I and project the guidance to E-space:
E, so(#;, σt) X
Tt gt
guidance in e-space
where X, is a (monotone) guidance schedule. We use a cosine decay d = Amax I+cos( with
7: E [0, 1] the normalized time (1 start, 0 at end).
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Algorithm 2 CL-DPS DPM (2M) (ours)
1: Imput: steps N, measurement y, Karras schedule {ar) pretrained encoder S. guidance weights ( #overlapping patches U.
2: Sample n (0, I); cache measurement patches {p]))=1.
3: Initialize ÆN
dummy prev. noise for warm-start
for N.N.1......
5: // denoiseer + CL guidance at (æ;,
6: Extract U overlapping patches {p) m. Jz=1 from æ;.
7: gi Vn, (((
[((x)))).
8:
E + Ea( 02) de 9.-
9:
_ Φ2(h) 1ー h; log log ot, $1 (h)
h²
10:
if N then
1;-1 0--
oi Di
00:1 - oi)
warm-start: 1-stage (IS) exponential Euler
12: else
13:
Ri-1 f "i
[0i(hs) or-1 88 + [0a(hs) 0.- (Ei+1 E.)
//DDM-Solver++ (2M)
14:
end if
15: Cache 3; for the next step.
end
17:
Return æzp.
Let {o N be decreasing Karras schedule with σy the start noise and do z O; define h, log 0:-1 log iz; DPM-Solver++<(2M) is second-order multi-step method for the diffusion ODE written log a time; it combines the current and one previous noise prediction. We use a warm-start (IS) step for the first interval, then 2M thereafter
As in the main text, we extract U overlapping patches of y once (cached), and of æt each step. Patch configuration and the overlap policy are identical to CL-DPS; only the state update (the sampler) changes.
Notes stability. (i) The E-space projection (o:gg) keeps units consistent with the denoiser output. (ii) We found a cosine-decay X; essential to avoid over-sharpening at low noise. (iii) For extremely strong guidance, a single medictor-correctto Heun sub-step large help, but was not required in our runs.
Unless specified, we N-50 steps, Karras p=7, and Amax E [0.5, 1.0] depending the operator family (same as the main CL-DPS).
F.1 WHY DPM-SOLVER++ IMPROVES OVER EULER/ANCESTRAL IN CL-DPS
Higher-order accuracy under external guidance. Let x()) follow the diffusion ODE in log-noise time, augmented with our contrastive guidance,
da
dlogg -o (Z) - \(o) a Valf (pl((patces((z)) (patches (y)))
Cl. guidance (m..y)
A first-order Fuler/ancestral step approximates this right-hand side constant over each interval, leading to O(h) local truncation error for step size log a;-1 log a; DPM-Solver++ (2M) second-order multi-step method that reuses the previous and current noise predictions, yiclding (h²) error while keeping the model interface unchanged. Because the CL-DPS term is just an additional drift, the same order improvement applies to the combined drift (denoiser guidance).
Stability at low noise and reduced variance. Near the end of the trajectory (a + 0), the iesasurmmenttuiddnne-tttttteeeee
ratio is largest; naive first-order updates overshoot produce ringing. DPM++'s multi-step correction better tracks the curvature of the drift, reducing late-step oscillations. Furthermore, when used in ODE mode (no extra noise injection), DPM++ avoids sample variance accumulation across steps, which empirically improves perceptual metrics (FID/LPIPS) without sacrificing fidelity.
We follow the common 2M recipe in public implementations; any equivalent coefficientization is acceptable.
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Table 4: Nonlinear blind inverse problems with stronger sampler. We insert DPM-Solver++ (2M)
into CL-DPS.
Rotation
FFHQ
(256
256)
AFHQ (256 x 256)
ImageNet
Method
PSNR
FID
LPIPS.
PSNR1
FID|
LPIPS.
PSNR
FID.
LPIPS_
CL-DES (SPE)
D (50)
22,98
31.85
0.295
21.79
34.52
0.312
20.31
43.14
0.333
CL-12P (JN1)
(50)
34.10
37.31
20.12
0.345
CL-DPS (SPE)
22.74
33.66
0,302
21.46
36,96
0.319
20.05
45.10
0.340
CL-DPS (JNI)
22.27
36.55
0.315
21.61
39.81
0,330
19.92
49.23
0,352
Zeom
FFHQ (256
256)
APHQ (256 x 256)
ImageNet
Method
PSNRT
FID
LPIPS]
PSNRT
FI1]
L.PISS
PSNRT
FID.
LPIPS_
CL-DFS (828) + DPM++ (50)
20.94
40.28
0.426
19.89
54.21
0.454
18.82
53.05
0.474
CL-DPS (JNI)
(50)
20.55
44.85
0.439
19.46
58.59
0.472
18.33
56.72
0.486
CL-DPS (SPE)
20.68
42.61
0.435
19.63
57.54
0.468
18.56
55.30
0.481
CL-DPS (JNI)
20.31
46.83
0.448
19.23
61.06
0.480
18.07
59.53
0.492
Schedule synergy. We discretize in log-or and adopt a Karras o-schedule, which allocates more steps to high-curvature regions of the flow. Together with simple cosine decay for A(a). this reduces discretization-guiiee mismatch and yields the small but consistent improvements reported below.
F.2 RESULTS WITH
(TABLE DESCRIPTION)
We replace the Euler/ancestral step in CL-DPS with DPM-Solver++ (2M) and keep all other components unchanged: same denoiser, same overlapping-patte encoder f. and cached measurement features. Unless noted, we use N=50 steps with Karras schedule (p=7) and cosine guidance decay X₂ from Amax
We keep measurement patches, stride/overlap, and the encoder backbone unchanged. Guidance is applied 6-space via the projection ē = ε > g to maintain unit consistency with the denoiser. reported numbers for the DPM++ rows were produced with the same number of function evaluations as the corresponding CL-DPS baselines.
Table 4 augments the main nonlinear blind deblurring benchmark by adding two rows-CL-DE (SPE) + +PM+ (50) and CL-DPS (UNI) + DPM+ (50) (50)-on the Rotation and Zoom tasks across FFHQ, AFHQ and ImageNet. The remaining are identical to Table in the main paper.
Across all datasets and both operator families, DPM++ yields: (i) slightly higher PSNR (typically +0.2-0.4 dB), (ii) lower FID (often -2 to -3), and (iii) lower LPIPS (roughly -0.005 to -0.015), with the largest gains ImageNet where the step budget is most constraining. This aligns with the reduced discretization error and improved late-stage stability discussed above. We emphasize that no retraining is required; the sampler swap is drop-in.
G IMPLEMENTATION DETAILS
G.1 TRAINING
Implementation details of training the auxiliary encoder. In all experiments, we use the ResNet-
18 (He et al. 2016) as the backbone model. We set the temperature 7 0.07 in Equation (T6), fix Y 256, MoCo queue length K = 65536, and momentum m 0.996. Also, we set patch size P-64 and stride 3-32 (50% overlap).
Auxiliary encoder training data construction. We train on aligned patch triplets constructed from clean images and synthetic measurements y = Ap(x) + c. At each step sample degradation family F E [Gaussian blur, motion blur, rotation blur, zoom blur) uniformly, then draw parameters from broad ranges and discard them afterward. Concretely, Gaussian blur standard deviation o Unif 2.4) pixels, motion blur uses length (ni 15) pixels and angle 0 Uni 180%), rotation biur uses angle & Unif(10", 30%) around random center,
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and zoom blur uses factor < 111, 1.6), average over 21 rotated/zoomed images. We add measurement noise E ~ (0, o with on ~ Unif(0.005, 0.03). For each spatial location we form query patch P.. from 2; its geometrically corresponding positive Py from y, and mixed pool of negatives Py drawn from other images, spatially mismatched locations, and a momentum queue. This yields the set Y used in the InfoNCE term of Equation (T6).
We train the model for 400 epochs, including a 5-epoch linear warm-up period, with the batch size of 256. Stochastic gradient descent with an initial learning rate of 0.03, weight decay of 0.0001, and cosine annealing is used for optimization.
We optimize the InfoNCE loss that aligns (o() with 94 (Pv) and repels all de (P,) in Y tempera- ture 7=0.07. The mixed negative construction is important, it prevents shortcuts based on low level statisties and improves invariance nuisance degradations while preserving sensitivity content. The MoCo queue of length K K665,530 536 provides stable and diverse negative set across steps.
For the momentum encoder, we set a momentum of 0.999. Data augmentation consists of random cropping with a scaling range of [0.045, 0.5) and an aspect ratio range of [0.5, 2] are applied to both {p(y|æ+)) Jezz and y. For each input image, we crop the given image at random location, and apply the color jitter as augmentation. For all inverse problems, Gaussian measurement noise with 0.02 is added (see Appendix for other noise levels). Full implementation details are available in our code repository.
For the FFHQ, AFHQ and ImageNet datasets, we utilize pretrained score functions following the configuration described in [hhng al. (2022b). To train the score function for kernels, we construct a dataset of 60k 64 x 64 kernels. Out of these, 50k motion blur kernels are produced using the implementation from [ where the blur intensity is sampled as I ~ Unif 1.0). The remaining 10k kernels are Gaussian blurs, generated with a standard deviation chosen randomly as (0.1,,0
Similar to the previous works (Chung et al. 2023a Laroche et al. 2024 Murata et al. 2023 Hamidi Yang 2024 Ye al. 2025). for FFHQ, we randomly select 50k images for training, and sample lk images of test data separately. For AFHQ, we train our model using the images in the dog category, which consists of about 5k images. Testing was performed with the held-out validation set of 500 images of the same category.
For the kernel and tilt-map score functions, we adopt the U-Net architecture provided in guided- diffusion [4] training the models under the default configuration.
G.2 FORWARD OPERATORS
We define forward operators for rotation blur and zoom blur that we use to synthesize measurements. Let x E HxxWxC be an image, c = ( (H-1 w-1) the rotation and zoom center, and let e ~
\O x denote measurement noise.
G.2.1 ROTATION BLUR
Given maximum shake angle 0, sample angles {M and nonnegative weights { that sum to I. The rotation blur averages rotated views around c:
M Σ
v(u)
Ro, (u) + E(uu,
(52)
where Rs applies rotation of 9 degrees about e with bilinear interpolation and reflect padding. For our experiments, we set M = 21, angles 0; linearly spaced in [-Ø, ø), sample weights wy oc d² exp( 203 with oe = 0/3, then normalized so that E, w = 1. The training range ø ~ Un0e 30%). Test ranges follow the benchmark specification. Interpolation is bilinear. Padding reflect. Channels are processed independently.
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G.2.2 ZOOM BLUR
Given zoom range [L, Smax], sample scale factors { and nonnegative weights ) that sum to 1. The zoom blur averages scaled versions about c:
M Σ
y(u)
wi Ze, x (u) + e(u),
(53)
where Z, scales by factor around For (minification) (magnification) we use bilinear interpolation. Padding is reflect.
area interpolation. For < 1
In the experiments we set M = 21, scales linearly spaced in [1, Smax]. Sample weights wi x
exp( (a. (a,-1)² with o. (Smax 1)/3, then normalized. We use the training range of Smax
2 Un1, 1.6). Test ranges follow the benchmark specification. Coordinates are centered at c. Interpolation choices above. Unless stated otherwise, we use 0.02 for the additive noise.
G.3 EVALUATION
We evaluate reconstruction quality using three standard metrics: Fréchet Inception Distance (FID) (Heusel al., 2017). Learned Perceptual Image Patch Similarity (LPIPS) (Zhang et al. 2018), and Peak Signal-to to- Noise Ratio (PSNR). Unless otherwise specified, we fix the number of diffusion steps to 1000 across experiments.
G.4 BENCHMARK METHODS
Pan-E_ Pan al. (017)) This method applies {_0 regularization jointly the image and kernel. We use the official codebase with the following hyper-parameters. Optimization and post-processing follow the same multi stage strategy as Pan-DCP.
A pixel= de-
grrad 4e 3 3
A_tv = le 3
A_1 = 2e 3
SelfDeblur Ren al. (2020). We adopt default YCbCr-based deblurring configuration. Training performed with constant learning rate of 0.01 for 2500 iterations. For the first 500 steps, optimization minimizes the MSE loss, after which it switches to minimizing 1 SSIM
DeblurGANv2 Kupyn et al. 2019). We adopt the official implementatior following the default hyper-parameter parameters, data augmentation strategies, and network design. Training minimizes a weighted combination of pixel loss, WGAN-gp adversarial loss, and perceptual loss, with as the generator backbone. Both FFHQ and AFHQ datasets are used, with training conducted for 1.5M iterations using a batch size of 1. As in MPRNet, the training set contains equal proportion of Gaussian and motion blurred images. The loss weights are:
A pixel 5e
Ladv = 3
A perceptuall 1e 2
H FURTHER STUDIES ON ENCODER CHOICE
Why CL-DPS (SPE) setup is meaningful? In many pipelines coarse family label available or can be obtained with negligible cost: (i) capture systems operate in discrete modes that are recorded as metadata, (ii) restoration stacks routinely gate inputs by a small set of operator families before
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invoking specialized solvers, (iii) when metadata is missing lightweight two-way classifier can detect the family with near- pear-perfect accuracy and tiny overhead compared to diffusion sampling. This keeps the setting blind to the unknown continuous parameters while matching practical deployments.
On the other hand, to train CL-DES DPS (UNI), the learned energy then estimates the mixture likelihood Pmiz æ;), which is exactly what is needed when the measurement family is unknown undetermined at test time. When the family A, is known, the same encoder can still be used with a simple restriction of negatives, but no change the loss required.
Let E O index the measurement family and xp denotes family-specific parameters. Given a clean image Mo, the forward model produces measurement
Aa ( + n2,
(54)
and let æy denote a diffusion state along the reverse process. During contrastive learning we first and then draw 1 p(gb O), where both To and p( O) are uniform
draw the family distributions. We then construct positives (æ,, y) from this mixture process, while negatives come
from large dictionary as discussed in Section 4.1
H.1 USING MAGHEER-CAPACTE ENCODER
To mitigate the performance gap introduced by CL-DPS (UNI), we train higher-capacity encoder namely ResNet-50, and evaluate it on the FFHQ dataset; results are reported in Table As seen there, scaling up the encoder model largely recovers the degradation caused by the mixture of the operators. We attribute this gain to the greater model capacity of the ResNet-50.
Table 5: CL-DPS (UNI) using model structures, namely ResNet-18 and ResNet-50.
FFHQ
Encoder
ResNet-18
ResNet-50
Distortion
PSNR
FID 4
LPIPS +
PSNR
FID 4
LPIPS
Rotation
22.27
36.55
0.315
22.60
33.95
0.309
Zoom
20.31
46.83
0.448
20.42
44.14
0.411
H.2 ROBUSTNESS TO FAMILY MISCLASSIFICATION
We simulate small fraction of inputs routed to the wrong encoder to approximate practical detector with imperfect accuracy. On FFHQ rotation we vary e E {0, 0.00,.0,0..8 0.10, 0.20) and report the PSNR, FID and LPIPS in the Table
Table 6: Numerical results robustness family misclassification
FFHQ
PSNR
FID
LPIPS
0
22.74
33.66
0.302
0.05
22.57
35.35
0.310
0.10
22.24
36.61
0.318
0.20
21.71
40.28
0.332
We observe near-linear degradation for small E. In internal test a two-way ResNet-18 family detector reaches 99.1% accuracy on held-out data with a runtime of about 1.1 ms per 256 x 256 image Nvidia H100 GPU. Diffusion sampling for 300 steps takes about 5.1 per image on the same GPU. The overhead of family detection therefore negligible relative to sampling time.
Knowing the measurement family yields the best quality. When the family is unknown, single cexture-trained encoder remains competitive and cross- fomsly usage still outperforms non-contrastive baselines. Small misclassification rates have modest effect quality in practice.
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1 DETAILS FOR THE TOY LIKELIHOOD GRADIENT CHECK
To quantify how well the surrogate guidance matches the true likelihood gradient in a case with a
closed-form V log p(y: ate), we generate Nwy 1100 grayscale images zo [0, 1|66x9 by summing 3 to Gaussian blobs, few straight edges, and Gaussian noise, then rescaling to [0, 1]. We use 1000 for training the auxiliary encoder and 100 for diagnostics.
For a parameter v we define a linear operator H applied to the vectorized image. Similar to the We sample & per image from of two families:
experimental setting in Section 5
Gaussian blur: s E [0.6, 2.0]. with kernel size of 5.
(55)
Motion blur:
intensity of [0.5 0.9], with kernel size of 5.
(56)
At diffusion step with noise level or we draw
Zt = -,30 + ofE, E~~ e~,,,,,1,
(57)
Y: = H xe Et, E N
(58)
The true gradient follows
ginue (1) = 2 (y:- H.).
(59)
We train a lightweight encoder f()) with an InfoNCE loss on pairs (Ix, yt)) drawn from the same forward model. Features are ng-normalized. Temperature is 0.07. We use a MoCo queue of K = 65536 negatives with momentum m = 0.996 and a 128-dimensional projection head. For diagnostics vary the dictionary size [Y] {64, 256, 1024) by sampling that many y!!) the same t. The surrogate is
(xp(( ())))))))))))))))))))))))))))( 1)/T)
ax.)) (e (+)
(60)
Σ3 exp( (J ()y())))/))
We compute
Jsur (t)
V æs log pog (3)
(t] xx.) V8+ En sy = (f(.)) (( (61)
Main experiments use 1000-step schedule. Diagnostics use subsampled grid Taiag of 50 indices that are equally spaced in log o., chosen from the same schedule. This preserves the early, mid, and late regimes with minimal clutter. We use batch size of 128, projection dimension 32, learning rate 1e-3, optimizer Adam with weight decay le-4, queue length 16384, momentum 0.996, temperature 0.07. Results robust to moderate variation of these values.
For each t Taiag and each Y we compute
(grnu (t), Isur (t)
I1 (t)
0: cos
I9rne (1)ll l| (1. (1)
Pe
1. (1)l22
(62)
averaging over 100 held-out images. We plot the per-t mean with a light moving average for readability. Raw curves and seeds are released with the code.
Under normalized features and a good encoder one expects 0, to decrease with t and or to approach one. Larger Y raises the positive softmax weight π and improves alignment. As check, increasing T or reducing Y degrades alignment as expected.
J CONVERGENCE BEHAVIOR OF CL-DPS
Here we study convergence under a nonlinear forward model on FFHQ. Figure we plot PSNR, FID, and LPIPS versus the number of denoising steps. The three baselines from prior work (BlindDPS, FastEM, and GibbsDDRM) do not improve with more steps, and their curves plateau or even drift, indicating that they fail to solve the underlying nonlinear inverse problem. In stark contrast, CL-DPS improves monotonically across all metrics and converges to substantially better reconstructions as the step count increases.
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302
18
300
203
234
Figure 5: Quantitative evaluation of PSNR, FID, and LPIPS across di diffusion steps for challenging nonlinear blind inverse problems. Top: rotation blur; bottom: zoom blur.
Table 7: Wall-clock runtime (per image, in seconds) of CL-DPS compared with diffusion-based baselines.
Method
FastEM
BlindDPS
LatentEM
GibbsDDRM
CL-DPS
Time (s)
46.03
47.68
46.87
51.85
90.84
K RUNTIME AND COMPUTATIONAL OVERHEAD
As discussed in the limitations, CL-DPS has higher runtime because it backpropagates through the encoder at every denoising step. We quantify this overhead by measuring per-image runtime on FFHQ with a single NVIDIA H100 GPU, using the same setting for all methods (1000 sampling steps, batch size 1). Wall-clock times for each method are reported in Table 7
Under this setup, CL-DPS takes 60.84 s per image, longer than the baselines Table 7 due to the extra backpropagations, but this additional cost enables CL-DPS to solve the non-linear inverse problems evaluated in our experiments, where the compared methods fail. In short, CL-DPS trades a modest increase in wall-clock time for substantially broader applicability.
L ADDITIONAL RESULTS ACROSS DIFFERENT NOISE LEVELS
To evaluate the robustness of CL-DPS under varying levels of measurement noise, we report additional results for both linear and nonlinear blind inverse problems across three Gaussian noise levels:
0.001, σ 0.02, and a 0.03.
Table 8 and Table (main paper) and Table present results for blind rotation and zoom deblurring on FFHQ and AFHQ datasets under 0.01, 0.02, and 0.03, respectively. As expected, increasing the noise level degrades the performance of all methods across PSNR, FID, and LPIPS. Nonetheless, CL-DPS consistently outperforms all baselines in both distortion and perceptual metrics, particularly in high-noise and nonlinear settings where competing methods struggle.
Similarly, to evaluate the linear blind inverse problems, we evaluate blind motion and Gaussian deblurring tasks under varying noise levels in Table 10 (a 0.01). Table 2 (main paper, = 0.02), and Table II (o = 0.03). CL-DPS again delivers competitive or superior performance across all 0.03, while other
metrics. Notably, CL-DPS maintains high reconstruction quality even methods suffer significant degradation. This highlights the robustness and generalization ability of contrastive-guided posterior sampling framework under blind measurement noise.
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Table 8: Nonlinear blind inverse problems o 0.01. Bold and underlined values denote best and second-best, respectively.
FTBQ (256 x 256)
AFHQ (256 x 2581
Method
Kotation
Zoom
Rotation
Zoom
PSNK
FID 1
LPISS
PSNR
FID
LPIPS
PSNK
FID 1
LPISS
PSNK
FID
LPIPS
CL-OPS (SPB)
24.35
28.93
0.274
22.12
38.21
0.412
22.85
33.50
0.295
21.05
53.45
0.445
310.2
260.1
TastEM
16.61
240.1
0.588
19.82
280.1
0.615
12.43
260.1
0.663
16.62
290.3
0.786
GibbDDRM
19.51
215.1
0.582
16.51
300.24
0.781
16.34
240.4
15.63
200.19
0.534
Table 9: Nonlinear blind inverse problems at o = 0.03. Bold and underlined values denote best and second-best respectively.
FFHQ
256)
APHQ
206)
Method
Rotation [TD
Zoom
Roiation
Zoom
PSNK
LPIPS
PSNR
TID
LPIPS
PSNK
F1D
LPIPS
PSNR
FID
LPIPS J
Ch-015 (8PF)
BlniDPS
16.31
165.4
0.568
15.54
120.5
0.867
12.75
215.4
0.891
11.25
290.1
0.634
FastEM
15.23
290.2
0.615
326.0
0.647
11.27
310.0
0.749
15.10
335.7
0.821
GibsDDRM
17.52
260.0
0.584
14.52
340.3
0.820
14.53
295.8
0.661
13.70
310.12
0.561
Table 10: Linear blind inverse problems 0.01. Bold and underlined values denote best and second-beet bestt respectively.
FFHQ (256 256)
AFHQ (256 x 250)
Method
Motion
Motion FID
Goussion FID..
PSNKT
FID.
T.IPSS
PSNK1
FIDL
LPIPS]
PENRI
L.IPSL
PSNK1
TPIPS_
DPS (SPE)
27.49
28.34
0.142
26.21
23.84
0.325
23.57
28.99
0.133
25.27
18.34
0.216
SeltDeblur
12.82
236.1
0.732
13.31
210.4
0.667
10.73
270.4
0.743
13.44
152.34
0.642
DeblurGANv2
19.33
180.2
0.541
21.14
155.3
0.511
18.75
155.6
0.565
22.25
74.24
0.495
Pan-ly
18.11
82.35
0.385
22. 22
75.76
0.361
17.88
205.2
0.603
23.92
56.35
0.365
BlindDPS
22.95
26.51
0.270
24.82
0.226
21.64
22.19
0.320
24.52
18.82
0,275
FasEEM
25.01
31.24
0.327
23.82
27.95
0.352
22.12
46.84
0.394
23.34
29.53
0.287
LatenDEMM
24.42
34.84
0.158
35.57
31,31
0.345
21.22
41.52
0.273
22.46
35.87
0.275
GiN+DDRM
26.65
36.25
0.150
27.34
30.85
0.410
22.96
44.24
0.172
24.43
39.23
0.384
Table 11: Linear blind inverse problems 0.03. Bold and underlined values denote best and second-best, respectively.
(256 256)
AFHQ (256 x 256)
Method
Motion FID.
Motion FIDJ
Gaussian
PSNR
I.PPSS)
PSNR1
FID
I.PIPS.]
PSNR
I.PISS)
PSNR1
FID.
I.PIPS_
24.36
38.26
23.03
31,06
0.373
20.81
38.88
0.235
22.62
24.54
0.255
SellDeblur
9.87
304.1
0.730
10.47
260.5
0.695
8.37
330,3
0.790
10.53
190.43
0.673
DeblurGANv2
18.36
Pan-ty
14.93
260.5
0.560
19.22
115.8
0.430
14.65
255.8
0.645
20.03
75.73
0.421
BlindDPS
21.52
34,53
32.42
0.251
19.92
29.63
0.360
22.64
23.54
FastEM
23.66
42.20
0.365
22. 16
37.52
0.390
20.84
62.04
0.335
22.63
40.45
0.317
LatenDDEM
21.97
45.76
0.385
19.79
54.47
0.302
21.27
42.54
0.305
GiNxDDRM
24.58
46.55
0.132
45.45
41.54
0.455
21 40
55.43
0.215
22.83
51.73
0,335
M ABLATION STUDY
M. ABLATION ON CONTRASTIVE HYPERPARAMETERS
We study sensitivity to the InfoNCE temperature т, the dictionary size |Y| used at guidance time, the MoCo queue length К during pretraining, the projection head dimension d, patch size P. and stride S.
The default parameters used the experiments in the main body of the paper are 7 = 0.07, Y = 256, K 65536, momentum ablation over all these hyperparameters. Reported numbers are averaged over Rotation and Zoom on the three benchmarks used in Table II
0.996, 128. P = 64 and S = 32. In the following, perform
Ablation Table 12 We observe that CL-DPS is relatively robust to the temperature 7 in the range 0.05 < < 0.10. A slightly lower temperature (r = 0.05) improves PSNR and FID on FFHQ and AFHQ, though it slightly worsens ImageNet fidelity (FII) rises by 0.5). Larger values
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Table 12: Sensitivity to temperature 7 in the InfoNCE loss for CL-DPS (SPE) setup.
FFHQ
AFHQ
ImageNet
PSNRT
FID
LPIPS
PSNRT
FID
LPIPS|
PSNRT
FID
LPIPS
0.05
22.88
33.12
0.298
21.61
36.43
0.316
20.38
45.64
0.344
0.07 (default)
22.74
33.66
0.302
21.46
36.96
0.319
20.45
45.10
0.340
0,10
22.67
34.25
0.306
21.32
37.55
0.323
20.36
46.02
0.345
0.15
22.41
35.11
0.312
21.15
38.49
0.329
20.12
46.98
0.352
(T = 0.15) consistently degrade quality across all datasets, confirming that overly smooth logits reduce the effectiveness of the contrastive guidance signal.
Table 13: Sensitivity to dictionary size |Y| used during guidance in CL-DPS denominator-awwe aware version only).
(SPE) setup (for the
FFHQ
AFHQ
ImageNet
PSNR1
FID
LPIPS
PSNR
FID4
LPIPS
PSNR
FID4
I.PISS
64
22.45
35.21
0.312
21.20
38.12
0.329
20.18
46.81
0.352
256 (default)
22.74
33.66
0.302
21.46
36.96
0.319
20.05
45.10
0.340
1024
22.59
32.79
0.296
21.66
35.89
0.313
20.62
44.17
0.337
Increasing the dictionary size |Y| improves all three metrics, with the largest gain seen when moving from Y = 64 to YY| 256. Gains from YY| 256 to | 1024 are smaller, suggesting diminishing returns. The trend most pronounced on ImageNet, where FID improves by more than point, indicating that richer dictionary better approximates the true likelihood gradient in high-diversity datasets.
Ablation on [Y], Table 13
Table 14: Sensitivity to MoCo queue length for CL-DPS
(SPE) setup.
K
FFHQ
AFHQ
ImageNet
PSNR
FID|
LPIPS
PSNR
FIDJ
LPIPS
PSNR
FID
LPIPSJ
8192
22.53
34.84
0.308
21.28
37.85
0.325
20.26
46.24
0.347
16384
22.61
34.33
0.306
21.36
37.28
0.322
20.33
45.83
0.343
65536 (default)
22.74
33.66
0.302
21.46
36.96
0.319
20.05
45.10
0.340
131072
22.69
33.59
0.321
21.56
36.42
0.317
20.42
44.81
0.338
Queue length K plays a similar role to dictionary size by providing harder negatives during pretraining. We observe consistent improvements as grows, with most notable jump between K 8192 and K 65536. The improvement saturates beyond K 65536, where increasing to K = 131072 yields only marginal additional benefit while incurring higher memory cost.
Ablation on K, Table 14
The projection dimension d also affects representation quality. Larger dimensions yield modest improvements across all benchmarks, but the relative gain between 128 and d = 256 is small compared to the additional computation and memory footprint. Hence, = 128 offers good trade-off between performance and efficiency for our default configuration. We show the visualization results in Figure 7
Ablation on d, Table 15
Ablation on P S, Table 16 This table demonstrates the effect patch size P and stride S on CL-DPS performance for both rotation and zoom blur. Reducing the stride from S-32 to S-16 (increasing overlap) consistently improves PSNR, lowers FID, and reduces perceptual error across all datasets, confirming that denser spatial coverage produces more stable guidance. Using smaller patch size (P=48) provides similar but slightly weaker gains, indicating that context loss offsets some benefits of denser sampling. Larger patches (P-96) or overlap (S=64) consistently
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Table 15: Sensitivity to projection head dimension d for DPS
(SPE) setup.
d
FFHQ
AFHQ
ImageNet
PSNRT
FIDI
LPIPS|
PSNR1
FID
LPIPS
PSNR
FID.
LPIPS,
64
22.55
34.51
0.307
21.56
37.51
0.324
20.28
45.94
0.346
128 (default)
22.74
33.66
0.302
21.46
36.96
0.319
20.45
45.10
0.340
256
22.75
33.69
0.299
21.58
36.89
0.316
20.55
45.17
0.339
Table 16: Sensitivity to patch size and stride. Default CL-DPS uses patch size P=64 and stride S=32 (50% overlap). Increasing overlap (smaller stride) consistently helps; removing overlap (S=64) hurts. We use CL-DPS (SPE) setup.
FFHQ (256 256)
AFHQ (256 X 256)
ImageNet
Rotation
PSNR1
FID.
IPIPS
PSNR1
FID
I.IPSS
PSNR1
FID..
LPIPS.
CL-DPS (P=64, S=32)
22.74
33.66
0.302
21.46
36.96
0.319
20.05
45.06
0.342
CL-DPS (P=64, S=16)
23.09
32.10
0.292
21.78
35.50
0.312
20.73
43.83
0.332
CL,DPS (P=48, $=16)
22.98
32.61
0.296
21.65
35.90
0.315
20.66
44.22
0.334
CL-DPS (P=96, S=48)
22.25
35.13
0.314
21.15
38.07
0.337
20.21
46.81
0.353
CL-DPS (P=64, S=64)
21.94
36.90
0.330
20.68
40.23
0.344
19.70
49.74
0.368
FFHQ (256
256)
AFHQ (256 X 256)
ImageNet
Zoom
PSNR1
FIDJ
LPIPS
PSNR1
FID
LPIPS
PSNR1
FIDL
LPIPS
CL-DPS (P=64, S=32)
20.68
42.61
0.435
19.63
57.54
0.468
18.56
55.30
0.481
CL-DPS (P=64, S=16)
21.12
40.21
0.422
20.15
54.34
0.451
18.95
52.90
0,466
CL-DPS (P=48, S=16)
21.05
40.80
0.424
20.02
55.03
0.458
18.86
53.80
0.472
CL-DPS (P=96, S=48)
20.32
44.56
0.444
19.34
58.67
0.482
18.35
56.80
0.498
CL,-DPS (P=64, S=64)
19.78
47.30
0.462
18.72
61.51
0.496
17.71
59.93
0.506
hurt reconstruction quality, especially on ImageNet, suggesting that overly coarse disjoint patch coverage fails to capture sufficient local detail.
We also provide example in Figure to show how the patchification works.
In addition, in order to show that how P & 5 values affect the quality of the restored images, we plot the images reconstructed by CL-DPS using different P & values. The results reported in Figure
M.2 ABLATION ON COL.RR CONSISTENCY HEAD
We qualitatively evaluate the effect of the color consistency head reconstruction quality. As shown in Figure S the model trained without the color consistency head (Figure ( produces a restored image with noticeable color shifts, particularly in the shirt region. In contrast, the model trained with the color consistency head (Figure (H)) recovers colors faithfully, resulting in visually consistent reconstruction that better matches the original image. This confirms that incorporating the color consistency head stabilizes the color distribution during training and prevents hue drift the restored outputs.
N
QUALITATIVE RESULTS ON ZOOM DEBLURRING
Zoom blur is among the most challenging nonlinear degradations for diffusion-based inverse solvers. often causing existing methods to produce severe artifacts or completely fail. As illustrated in Figure 12 benchmark methods such as BlindDPS, FastEM, and GibbsDDRM struggle to recover fine contrast, CL-DPS reconstructs a visually coherent image with accurate structure and color, demonstrating its robustness under extreme nonlinear conditions. These results highlight that CL-DPS is not merely competitive but uniquely capable handling severe zoom blur without catastrophic failure.
details and exhibit strong distortions.
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Figure 6: Example of patchified image of resolution 256 x 256, with stride size of 64 and patch
size of 128 128.
(a)
(b)
(c)
(d)
Figure 7: Ablation study of different inference strategies. (a) Original image, (b) Measurement, (c) Restored image with stride 64, patch size 32, (d) Stride 256, patch size 256, (e) Stride 64, patch size 64, (f) Stride 48, patch size 16.
QUALITATIVE RESULTS ON LINEAR DEBLURRING TASK
Gaussian blur and motion blur are two well-studied cases in blind diffusion- based deblurring. To demonstrate CL-DPS's superiority on the blind linear deblurring task, we present additional visual results and compare reconstructions against two baselines, BlindDPS (Chung et al. 2023a) and
(a)
(b)
(c)
(d)
Figure 8: Ablation study the color consistency head. (a) Original image, (b) measurement, (c) restoration without the color consistency head (color shift visible), (d) restoration with the color consistency head (color faithfully preserved).
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(a)
(b)
(c)
(d)
(l)
Figure 9: Qualitative results on blind zoom deblurring, challenging nonlinear inverse problem. (a)
Ground truth image, (b) zoom-blurred measurement, and restorations from (c) BlindDPS (Chung
et al. 2023a). (d) FastEM (Laroche et al. 2024), (e) GibbsDDRM (Murata et al. 2023). and (f)
CL-DPS (ours). Competing methods fail catastrophically, whereas CL-DPS successfully reconstructs sharp and color-consistent image.
(a)
(b)
(c)
(d)
Figure 10: Blind Gaussian deblurring results: (a) ground-truth image, (b) caussian-blureed measure- ment, and restorations using (c) BlindDPS (Chung et al., 2023a). (d) GibbsDDRM (Murata et al. 2023 and (e) CL-DPS (ours). Visually, CL-DPS produces more natural images.
GibbsDDRM (Murata et al. 2023). Results for Gaussian blur and motion blur are shown in Figure I0 and Figure [1 respectively.
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(a)
(b)
(c)
(d)
(e)
Figure 11: Blind motion deblurring results: (a) ground-truth image, (b) botured measurement, and restorations using (c) BlindDPS (Chung al. 2023a), (d) GibbsDDRM (Murata al. 2023).
and (c) CL-DPS (ours). Visually, CL-DPS produces more natural images.
T=400
T=600
T=800
T=1000
Measurement
T=200
T=100
T=50
T=0
Ground Truth
Figure 12: The CL-DPS process of recovering the zoom blurred measurement.
P DENOISING PROCESS OF CL-DPS
Here, we visualize the denoising process of CL-DPS over 1000 timesteps. To this end, we select a single image and display the reconstructed images throughout the denoising process, illustrated in Figure [2
32


Reviews:
[{"review_id": "HkC1YHROg5", "reviewer": "Reviewer_s2GW", "summary": "The paper introduces CL-DPS, a new diffusion-model framework for blind nonlinear inverse problems, removing the need to know measurement operators at inference. The method uses a contrastive learning trained encoder to approximate the conditional likelihood on some synthetic dataset. At test-time, the encoder is used to approximate the likelihood $p(y |x_t)$ and guide the diffusion sampling without operator inversion. The experiments show CL-DPS outperforms existing methods on complex nonlinear tasks and is competitive on linear benchmarks.", "questions": "1. How well does the method generalize to real-world, non-synthetic degradations?\n\n2. Could the method be applied to more complex degradations such as unstructured degradations like rain for example?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. The method uses a synthetic dataset to train the encoder making it still a blind method which is practical.\n\n2. The method outperforms the compared methods on different tasks.", "weaknesses": "1. The method requires some training, which increases the computational cost compared to other zero-shot methods that are training-free.\n\n2. I have concerns about the fairness of the evaluation. Basically you are training your contrastive encoder on some degradations and then testing on them while the other methods do not. How do you think that this evaluation is conclusive on the superiority of the method given that the other methods do not have this advantage. A comparison with methods that use a synthetic dataset for training is needed.\n\n3. To me, to prove that the method is really interesting, it should be shown that it works better than the existing training methods in the out-of-distribution case. Otherwise, I do not see a big advantage compared to simply learning the degradation operator using the synthetic dataset.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "TKdiyxyuE1", "reviewer": "Reviewer_unhN", "summary": "This paper proposes a diffusion-based framework for solving nonlinear blind inverse problems, a setting that remains largely underexplored in the current literature.\nThe authors introduce a MoCo-style auxiliary encoder, which is ingeniously integrated into the diffusion inference process to address the challenges brought by both nonlinearity and blindness in the observation model.\nExperimental results demonstrate that the proposed method can effectively handle complex nonlinear blind inverse problems that existing diffusion-based approaches fail to address, while maintaining competitive performance on standard linear blind inverse benchmarks.", "questions": "What is the generalization ability of CL-DPS? Does it require training a specific auxiliary encoder for each dataset, or can a single encoder generalize across different data domains?\n\nI also have a question about the input used during the training of the auxiliary encoder.\nThe goal of the auxiliary encoder is to cluster samples generated by the same blur kernel, which is not entirely consistent with MoCo’s objective of grouping semantically similar samples.\nFor example, how does the method avoid clustering together samples generated from the same x_t but under different blur kernels, given that these samples may appear very similar in feature space?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1.\tThe paper addresses a novel and underexplored problem—nonlinear blind inverse modeling—and presents, to the best of my knowledge, the first diffusion-based solution to this setting.\n2.\tThe proposed approach is conceptually sound and empirically effective, as demonstrated through comprehensive experiments.\n3.\tThe writing is clear and well-structured: the motivation, background, and related work are coherently presented, and the proposed method is easy to follow.\nImportantly, the authors support their claims with both theoretical analysis and solid empirical validation.", "weaknesses": "1. The main contribution of this paper lies in the design of the auxiliary encoder, but some important details remain unclear. For example, how is the training dataset for this encoder constructed?\n2. The proposed CL-DPS increases the runtime by about 20% compared to the baseline. However, considering that CL-DPS can handle nonlinear blind inverse problems that the baseline methods cannot, this increase is acceptable.\n3. The setup of the baselines requires further clarification. Since methods such as BlindDPS rely on the assumption of a linear observation model, it is important to explain what modifications were made so that these baselines can handle nonlinear cases while ensuring a fair comparison.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "bFTjxtGyNb", "reviewer": "Reviewer_c2Zw", "summary": "This paper introduces a framework for solving blind nonlinear inverse problems where existing diffusion models fail. It works by training an auxiliary contrastive encoder offline to learn a likelihood surrogate, which is then used as gradient guidance during diffusion posterior sampling to restore images without knowing the specific degradation operator. The evaluations are comprehensive.", "questions": "See weaknesses.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "1. The paper manages to apply diffusion models to the blind nonlinear inverse problem, a domain where previous SOTA DM-based methods often fail.\n\n2. The core idea of using an offline, MoCo-trained auxiliary encoder to learn an amortized likelihood surrogate over a distribution of operators is reasonable and novel.\n\n3. The empirical results are comprehensive. CL-DPS produces high-quality restorations on challenging nonlinear tasks (rotational/zoom deblurring).", "weaknesses": "1. Inference Cost: The primary weakness is computational overhead. The method requires a forward and backward pass through the auxiliary encoder at every sampling step to compute the guidance gradient. This is a significant practical limitation. Can the cost be further reduced?\n\n2. Reliance on Special Encoders: The best-performing model requires training a separate encoder for each family of operators. This assumes the operator class is known at test time, which is a strong assumption that weakens the \"fully blind\" claim. The fully blind UNI model shows a consistent performance drop. While Appendix H.1 shows that a larger ResNet-50 can close this gap, this trade-off between generality and performance/efficiency is a key weakness.\n\n3. Mismatch in Guidance: The main algorithm uses a simplified \"numerator-only\" guidance gradient, which deviates from the full, theoretically-derived gradient. While Appendix E provides an empirical justification (showing the denominator term is small and its inclusion gives minimal benefit for high cost), this is a disconnect between the presented theory and the practical implementation.\n\n4. How does the model perform on OOD parameters, such as a $40^{\\circ}$ rotation blur? Does it degrade gracefully or fail abruptly?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "kIAs5cWVb9", "reviewer": "Reviewer_qhCf", "summary": "Posterior sampling with diffusion models requires computing the gradient term $\\nabla_{x_t} \\log p(y \\mid x_t)$ when solving the reverse ODE. To estimate this term, the paper trains an encoder $f$ via contrastive learning, such that the log-likelihood gradient can be approximated by the gradient of a contrastive score - defined as the inner product between $f(x_t)$ and $f(y)$. By training the encoder on synthetic measurements, the method enables blind inverse problem solving for both linear and nonlinear forward operators. Extensive experiments on Gaussian, motion, rotational, and zoom deblurring tasks demonstrate that the proposed approach outperforms existing baseline models in reconstruction quality.", "questions": "How the authors applied BlindDPS for zoom and rotation blur? As diffusion prior for blur kernel is only trained for the gaussian and motion blur, it is not straight-forward to use it for those tasks.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper proposes a way to solve non-linear blind inverse problem, which is not actively explored with existing baseline method, yet.\n- The paper provides various empirical analysis on effects of introduced components (e.g. patch-wise inference, regularization, dictionary size...) as well as theoretical analysis. \n- The proposed method is more effective than baseline methods.", "weaknesses": "- Problem is partially blind: encoders are trained for a family of measurement, but with limited number of factors (e.g. two known angles for rotation blur).\n- Missing details for the forward operations. Especially, there is not sufficient information for implementing zoom and rotation blur.\n- Missing related works [1], which is originally proposed for linear inverse problem, but could be used for non-linear inverse problem too.\n\n\nReference \n\n[1] Diffusion Prior-Based Amortized Variational Inference for Noisy Inverse Problems, ECCV2024", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)