Taming Score-Based Denoisers in ADMM: A Convergent Plug-and-Play Framework
Authors: Rajesh Shrestha
Xiao Fu
Venue: ICLR 2026 Poster (2026)
Keywords: diffusion
score model
inverse problem
convergence
optimization
generative model
Primary area: generative models

Abstract: While score-based generative models have emerged as powerful priors for solving inverse problems, directly integrating them into optimization algorithms such as ADMM remains nontrivial. Two central challenges arise: i) the mismatch between the noisy data manifolds used to train the score functions and the geometry of ADMM iterates, especially due to the influence of dual variables, and ii) the lack of convergence understanding when ADMM is equipped with score-based denoisers. To address the manifold mismatch issue, we propose ADMM plug-and-play (ADMM-PnP) with the AC-DC denoiser, a new framework that embeds a three-stage denoiser into ADMM: (1) auto-correction (AC) via additive Gaussian noise, (2) directional correction (DC) using conditional Langevin dynamics, and (3) score-based denoising. In terms of convergence, we establish two results: first, under proper denoiser parameters, each ADMM iteration is a weakly nonexpansive operator, ensuring high-probability fixed-point *ball convergence* using a constant step size; second, under more relaxed conditions, the AC-DC denoiser is a bounded denoiser, which leads to convergence under an adaptive step size schedule. Experiments on a range of inverse problems demonstrate that our method consistently improves solution quality over a variety of baselines.

Published as a conference paper at ICLR 2026
TAMING SCORE-BASED DENOISERS IN ADMM: A CONVERGENT PLUG-AND- PLAY FRAMEWORK
Rajesh Shrestha
Xiao Fu School of EECS Oregon State University xiao. www.rrgonsttteeecoommmm
School EECS
Oregon State University
shrresthf@orgoonsttee.eee
ABSTRACT
While score-based generative models have emerged as powerful priors for solving inverse problems, directly integrating them into optimization algorithms such ADMM remains nontrivial. Two central challenges arise: i) the mismatch be- tween the noisy data manifolds used to train the score functions and the geometry of ADMM iterates, especially due to the influence of dual variables, and ii) the lack of convergence understanding when ADMM equipped with score-based denoisers. To address the manifold mismatch issue, we propose ADMM plug-and- play (ADMM-PnP) with the AC-DC denoiser, a new framework that embeds a three-stage denoiser into ADMM: (1) auto-correction (AC) via additive Gaussian noise, (2) directional correction (DC) using conditional Langevin dynamics, and (3) score-based denoising. In terms of convergence, we establish two results: first, under proper denoiser parameters, each ADMM iteration is a weakly nonexpansive operator, ensuring high-probabilltty fixed-point ball convergence using constant step size; second, under more relaxed conditions, the AC-DC denoiser is a bounded denoiser, which leads convergence under an adaptive step size schedule. Exper- iments on range of inverse problems demonstrate that our method consistently improves solution quality over a variety of baselines.
INTRODUCTION
Inverse problems arise in many fields, including medical imaging (Song et al. 2022b; Jin et al. 2017]
Arridge 1999), remote sensing (Entekhabi et al. [994 Combal et al. 2003). oceanography (Bennett
1992). and computational physics (Raissi al. 2019 Tarantola| 2005). Their solutions typically rely
on incorporating prior knowledge or structural assumptions about the target signals, either through explicit regularization or data-driven models.
Classical approaches to inverse problems often rely on handcrafted regularizers, such as the & norm for sparsity (Yang al. 2010 Elad & Aharon 2006 Dabov et al. (007) and the nuclear norm for low-rank structure (Semerci et al.) 2014 Hu et al. al. (017)) Deep learning introduced a new paradigm of using learned generative models VAEs, GANs, and normalizing flows-as data-driven regularizers (Ulyanov et al.) 2020 Alkhouri et al.] [2024] Shah & Hegde, 2018), offering more expressive priors by capturing complex distributions. More recently, pre-trained score functions from diffusion models have gained attention for inverse problems (Song et al., 2022b) Chung et al. 2023), as they effectively approximate data distributions and align solutions with the underlying data geometry (Xiao et al. 2022).
The use of pre-trained score functions in inverse problems mainly falls into two categories. The first modifies the MCMC process of diffusion sampling to incorporate observation information, as in DPS (Chung et al. 2023) and DDRM (Kawar et al. 2022), where observations guide unconditional score functions to perform posterior sampling. The second integrates score functions into deterministic or stochastic optimization algorithms; for example, Wang et al. (2024) and Song et al. (2023) use them as "projectors" to keep iterates the desired data manifold. Furthermore, building Tweedie's s lemma, which links score functions to signal denoising, works such as (Zhu et al., 2023 Mardani
al. 2024 Renaud et al. 2024b) employ score denoisers in proximal-gradient steps.

---

Published as a conference paper at ICLR 2026
Challenges. The works in (Zhu et al. 2023 Mardani et al. 2024 Li et al. 2024 present flexible "plug- and-play (PnP)" paradigms that integrate diffusion models with optimization algorithms. However, two challenges remain in this line of work. First, score functions trained noisy data manifolds constructed via Gaussian perturbations, whereas optimization iterates need not lie on such manifolds, leading geometry mismatch and degraded denoising performance. Remedies such stochastic regularization (Renaud et al. (2024b) or purification (Li et al. 2024) add Gaussian noise to the iterates, but this does not guarantee alignment with the score manifolds. Second, the theoretical understanding of these methods frrthodssppaticlaaar particularly their convergence properties when combining the score denoisers with various optimization paradigms- paradigms-remains remains limited.
Contributions. We propose to integrate score- based denoisers with the ADMM framework. Using ADMM iterates with score-based denoising particularly challenging, as the presence dual variables further distorts the "noise" geometry-likely explaining why score-based denoising has rarely been combined with primal-dua methods. Nevertheless, ADMM remains attractive for its flexibility in handling diverse inverse problems with multiple regularizers. Our contributions are:
Score-Based AC-DC Denoiser: To mitigate the manifold mismatches, we propose three-stage denoiser consisting of (1) additive Gaussian noise aulo-correction (AC), (2) conditional Langevin dynamics-based directional correction (DC), and (3) score-based denoising. The AC stage pulls ADMM iterates toward neighborhoods of noise-trained manifolds, while DC refines aligmment without losing signal information. This combination balances efficiency and accuracy, making score-based denoising effective within ADMM.
Convergence Analysis: We show that, under proper AC-DC parameters, each ADMM iteration weakly nonexpansive, ensuring convergence to a fixed-point neighborhood under constant step sizes under strongly convex losses. We further relax convexity and prove that an adaptive step-size scheme (Chan al. 2016) guarantees convergence with high probability. These results extend prior ADMM-PnP convergence theory (Ryu et al., 2019 Chan et al. 2016) to ccore-based settings.
Our method validated on diverse applications includiing inpainting, phase retrieval, Gaussian and motion deblurring, super-resolution, and high dynamic range (HDR).
Notation. The detailed notation designation is listed in Appendix A.1
BACKGROUND
Inverse Problems. We consider the typical inverse problem setting where
y = A(x)= +
(I)
where A: Rd R is the measurement operator, S. d, and & is additive noise. In some inverse problems, e.g., signal denoising and image deblurring, we have n = d; while for some other problems, e.g., data compression and recovery, have d. The goal is recover from the y, with the knowledge of A. Structural regularization on æ is often used to underpin the desired solution:
min l(y || A(x)) h(x).
(2)
where f(y || A(x)) is divergence term that measures the similarity of y and A(x) (e.g., l Alx)])3, and h(x) is a structural regularization term (c.g., ||ز||| for sparse σ).
Diflusion-Based Inverse Problem Solving. Diffusion models can also be used for solving inverse problems, in ways more subtle than direct regularization. Consider training a diffusion model on To Pdata via denoising score matching (Song al.] (021). where the forward process is 2,120 ~ N (t) t E [O, with variance schedule a()) = and (t)) increasing in t. After training, the model provides score function s0(ar, (() V.,, logp(a;; where p(a:.) the marginal density of a, This induces noisy data manifolds
Mo(t) supp(x:), Vt T
which are continuous since æ; is generated by Gaussian perturbations of 2n ~ Pdata- These score functions then be leveraged in different ways assist inverse problem solving.
Posterior Sampling: Many works formulate inverse problems as posterior sampling from p( y) oc p(z)p(yja æ²) These methods approximate Va, log p(y) ax.) and combine it with the learned score
2

---

Published as a conference paper at ICLR 2026
(a (t) V log to perform stochastic sampling t decreases (Chung et al. 2023 Song eet al. 2022a Kawar et al. 2021 2022 Wang et al. (2023). While effective, their performance is often limited by the accuracy of the approximation to V pzy)(z)
Plug-and-Pae Play (PnP) Approaches: Instead of sampling schemes, another line of work employs deterministic or stochastic optimization to solve 2 pllugiig score functions into the updates as
structural regularizers. A representative example is DiffPIR (Zhu et al. 2023) which adopts a
variable-splitingg reformulation, minz, = ((z)))((z=)) + = 2 h(z), where the z-subproblem at iteration & reduces to a standard denoising step:
(k+1) arg min 11
x (R+1) - 2 =|² + +z()..
Denoising:
(3)
2
This step is then tackled using a score-based denoiser:
2(++1) x Daw (xx(+1)) = (k+1)) + (o(k))) (++11, ((k)
(4)
where (k++1) x (k+1)
+ o (()) (((00 (((1")"""11)))/ 1/00))
()) with ~ NO, I) and Ç [0, 1]; (1 +1
2024) for a similar method. These denoisers are designed following the Tweedie's 's also see (Li et al.
(A). The construction of (*)) is meant to make the inputs to the score function closer to certain (M) Using score-hased denoising, h(2)) is implicitly reflected in the denoising process and thus does not need to be specified analytically.
lemma Robbins
1992) (see Appendix
Another line of approaches explicitly construct regularizers h((= whose gradients correspond to applying the score function. Examples include RED-diff (Mardani et al. 2024) and SNORE (Renaud)
al. 2024a). In SNORE, the regularizer is defined h(()) Ezj log Per (3:), < x ae,
() and taking its gradient in 2 yields:
SNORE update: (k+ x (R)
oVE(y]||A(a?))) - ()()(()) - Dar (x)
(5)
where, again, Dz मे - 0²80 (z. or by the Tweedie's lemma.
Challenges- Manifold Mismatch and Convergence. Score-based PnP methods face two main challenges. First, the score is not trained on algorithm-induced iterates (c.g.. (k+1) in 3). While
both z, and (A) can be seen as noisy versions of æ ~ Pdatas Iy follows Gaussian noise whereas the distribution of (k) is unclear. Many works attempt to bridge this gap by injecting Gaussian noise before applying the score function (cf. T (k+1) in (3 (k) in 5), or by purification-bast schemes (Nie et al. 2022] AAlkhouri et al., 2023 Meng et al. 2022). Yet noise injection alone is
insufficient, and overfitting to measurement noise remains an issue (Wang et al. 2024). Second,
the understanding to convergence of score-based PnP remains limited. Unlike classical denoisers with established theory, the geometry mismatch above makes it unclear whether iterates stabilize or under what conditions convergence can be ensured. Existing analyses mostly cover primal algorithms (see, e.g.. (Renaud et al. 2024a)). Primal-dus methods such as ADMM offer greater flexibility for handling multiple regularizers and constraints, but their convergence with score-based denoisers remains unclear, as dual variables further complicate the manifold geometry of the iterates.
PROPOSED APPROACH
In this section, we propose our based denoiser and embed it in the ADMM framework. While we focus on ADMM due to its flexibility, the denoiser can be plugged into any other proximal operator based schemes (e.g. proximal gradient variable-splittnng as in DiffPIR).
Preliminaries of ADMM-PnP. ADMM-based inverse problem solvers start by rewriting 2 as
min ((y [A(=)) + yh(z) s.t.
(6)
The augmented Lagrangian of 6 is given by L, (x, y, y; = f(y) [A(x)) + +h)=)+++++((() 2) + f >1|2 where À is the dual variable and O is the penalty parameter. At iteration k, ADMM updates are as follows:
an (k+1 arg min 1 (((gllA(x))) + + (kk) (k)
(7a)
m P
2
0 2 x (k++) + (x) 2
2 (++1) arrgmin min _1(z)
(7b)
(*+1)) (t) (z) (k+11) -2(++1)
(7c)
3

---

Published as a conference paper at ICLR 2026
g())
Zuc
(2100)
(K)
Maata
M o0:
Mato
M data
M
Mato
Sg trained over data from (Mores)oo
Figure 1: Left: direct denoising (()) using score functions could lead to unnatural recovered signals with artifacts. Right: AC-DC denoising brings ((() closer to Mage), and then uses the score function to bring (kk the data manifold Maata-
Algorithm 1 AC-DC Denoiser at iteration A of ADMM in 7
I: auto correction (AC): zac < g(() + (1, n ~ I)
2: directional correction (DC):
3: (()) + 2nc
(k)
4: for to do
5:
(k.++11) (k.5)) (k) (Vo2 (=l)) (k + (wo(t.)) ,(())))))))))) + (m)m)
end
7: Zde
(K)
(J)
8: Denoising:
Z: + B[zo = ac
= zae (1 1 (a0))
(Tweedie Denoising) (ODE Denoising)
(Alternative: Rdde + 20 by solving:
(k)
del dt
A(t)so(zee,)) t), with 2.(4) (K)
where (R) X (k) is the scaled dual variable. The subproblem (bb is denoising problem, and thus can be replaced by
(++1) Prox (k++1) + (R)) = Doce (Z(1)).
The above is the classical ADMM-PuP method; see (Chan et al. 2016 Ryu et al. 2019). Same before, Eq. 8 can be replaced by the score-based denoising following the Tweedie's lemma. However, z (K) = (x(++1) + (4) could be in any of the manifolds Ma(t)) which the score function was trained, such naive replacement does not ensure effective denoising. The existence of
the dual variable (*) makes the noise distribution in (A) even harder to understand.
Proposed Approach: The AC-DC Denoiser. To address the manifold mismatch issues, we propose a three-stage denoiser. To be specific, in the kth iteration of the ADMM algorithm, we use the denoising process shown in Algorithm TT. Note that the Tweedie's lemma step (line 8) can also be substituted by score ODE based process Karras al. 2022 initialized at = Our algorithm using these two different denoisers will be referred to as Our twediie and Ours-ode, respectively.
(k)
The rationale of the AC-DC denoiser is illustrated in Fig Recall that the score function is most effective on the noisy data manifolds [MMot) as it is trained over them. Since ADMM-induced
iterates (kk need not lie these manifolds, directly applying score-based denoising may be ineffective. The AC step addresses this by adding Gaussian noise, making z closer to some Marti (see Appendix B). This idea is related to the "purification" step in (Nie et al. 2022 Alkhouri et al. Li et al. 2024) and noise-added denoising in (Mardani et al,) 2024] Renaud et al., 2022a]] /Zhu 2023
2023
(cf. and (3)) However, AC alone does not guarantee manifold alignment. The proposed DC step, based on Langevin dynamics, further refines z toward M,
al.
To see the idea, let break down the three steps. First, the AC step gives
(k) =.(k) + (()) 8 ((k)
(k)
(k)
Zac
= 2 Pdata
4

---

Published as a conference paper at ICLR 2026
(k) (1)) 8(k) (2 n2 + s(),, and where 2y (8) is denoised signal of 2(8), (x( 2
R1, n2 lid (O Ia). Given sufficiently large o (b*) zac would have dominated by Gaussian noise- but not necessarily on any of Mo, where the score was trained. Starting from zac the DC step runs a few iterations of Langevin dynamics targeting the distribution p(z=(t) |=(()). This is because supp(=,,<)) (xae)) S supp(zz,e)) (M) In addition, (())(() (kR) the same time retains the information of Zac (thereby the information from the measurements). Assume that the forward process used in training the score has sufficiently small time intervals, Ma( is approximately contained in (M,))==- This way, when applying Tweedie's lemma for denoising, the step is expected to be effective, the score was trained over (M
(A)
Note that the conditional score for the Langevin dynamics step can be expressed as follows
V log (200 = s0 (0(2 (b))) + V Vlog((>
(=)()))- 1=x()))
(10)
Ideally, one would use the exact V log p( Zac (k) practice, we approximate p(=nc (0) using Gaussian distribution. Note that under proper scheduling of (k) and mild regularity conditions on s(*), e.g., when Var(s(k)) < o(k), the likelihood can be well-approximated by a locally quadratic form, leading to V Flogp(zz (z0()) R
(a) ) for the DC step-whichh which is unavailable. In
(k)
11/03 (Eg()) (k)) and the DC step in Algorithm
4
CONVERGENCE ANALYSIS
4.1 CONVERGENCE OF UNDER WEAKLY NON-EXPANSIVE RESIDUALS
Following the established convention in ADMM-PnP, e.g., Buzzard et al. 2018 Sun al. 2019 Chan 2019 Teodoro et al. 2017), we aim at understanding the convergence properties when the AC-DC denoiser is used. We will use the following definitions:
Definition 1 (Fixed point convergence). Let T X X be the update map of an iterative algorithm, and let (0) E X be arbitrury initializution. The algorithm is said to converge to fixed point xt if for any > there exists K5 > such that the sequence generated by the algorithm (x(1)) IkeNt
satisfies (xx( - 21112 < 8 for all k 2 Ks. Equivalently, limg-+00 x(k) = xx with T(x*) = =x"
Definition 2 (Sequence convergence a 8-ball). For a certain ō > O, a sequence ( to converge within a 8-ball if there exists K >> O and x² such that the following holds for all k 2 K.
JKEN+ is said
(x)
x 58
(11)
weaker statement; that is, even when k + oc, (*k) - xx does not necessarily happen. Nonetheless, convergence 5-ball is still meaningful. The notion of ó-ball convergence is often used in numerical analysis for stability characterization:
Comparing the two definitions, Definition
see, e.g.. & Argyros 202T); Ren & Wu (2009) Liing (007).
Definition 3 (ADMM convergence to a 8-ball). ADMM is said to converge within 6 ball if the sequences (x) and (u(*)) obtained from ADMM converges within 8-ball.
To proceed, consider the following assumption:
Assumption 1. For a certain 5 > there exists such that for all ž 22 E Rd. the following holds:
|| (22)) / -
- 82
(12)
where RS (2) (Dom) p()) [2) with I being the identity function (i.e., [(z) 2).
Here, the notation (Do (z) denotes the residual of D Deceo i.c. D extends the fixed point convergence of ADMM- Ryu et al. 2019). Unlike Ryu et al. (2019) where R5 needs to be strictly contractive, our result allows R2 to be weakly contractive:
Dec ( (2) Σ. The next theorem
Theorem 1. Under Assumption ! assume that & p-strongly convex. Then, there exists x", u" and (
K > 0 such that the sequences x ())) JEEN and {u ()) keNt generated by ADMM-PuP using a fixed
5

---

Published as a conference paper at ICLR 2026
step size p satisfies ||((() u²l12 and |xx00] -x" x011 S with (1 + P 1 2 for all
k N K when +/u(++)-22)) < 1/p where 8 8' and p 2nc²
p+p+2pp
The proof is relegated to Appendix C
Note that when 5 = 0, it implies the result in Ryu et al. (2019).
4.2 CONVERGENCE UNDER WEARLY NON-EXPANSIVENESS WITH AC-DC
In this subsection, will show that the AC-DC denoiser satisfies Assumption Junder mild conditions. To this end, consider the following:
Assumption 2 (Smoothness of log Pdata). The log data density log Pdata M-smooth for a constant M > 0, i.e. V log Pdata (x) - V log Paata (B)ll2 S M || V1l2 for all E X.
Assumption (Coercivity for Pdata)) There exists constants e₁ > 0 and C2 2 0 such that
V log (atta(a)) 2 C1 log Pdata
c2, || S -c log (data (x)) c2; Vx X
(13)
This coereivity assumption means the negative log-density grows sufficiently fast at infinity, which prevents the Langevin dynamics from "escaping to infinity". This assumption guarantees stabillity and ensure ergodicity leading to convergence to the stationary distribution (Mattingly et al.] 2002).
Theorem 2. Suppose that the assumptions in Theorem 0
Assumption 2 and Assumption 3 hold. Further, assume that the DC step reaches the stationary distribution for each k. Let D Daw
(1) denote the AC-DC denoiser. Then, have:
zoo
(a) With probability at least - -2e- "*, the following holds for iteration of ADMM-PPP:
(I(Dam) - (z) (Domn - 1)())l1 S c - -y12 + 8%
(14)
for any y, E X and k: E N+ when (g00) + (o(1)) 1/M with
& = 3( 3((V/M@210/11d M)² + (a (x2
(15)
o = 3(2 )3 + du + 2vx)= + 32d0 (ore" Mo" 200) ) log 2lgg)
(16)
In other words, with v In 27/61 + In k, the denoiser Dyw, satisfies part for all k N+ with probability at least - 17.
(b) Assume that (80 is scheduled such that lim (k)))) = 0 v2 In 25/6n + 2 1. and c/p(1+c-2+?) < 1/p all hold, where c = lim sup E2 with e defined in (75). Consequently, 5 = limg-+ac sup 8% is finite and ADMM-Phl with the AC-DC denoiser converges to an r-ball (see in Theorem 7 with probability at least 1 n.
The proof is relegated Appendix D Theorem 2 establishes that the AD-DC denoiser is weakly non-expansive with probability 1 1 20---== * in iteration k. The (b) part states that when (b))) is carefully scheduled to approach zero as grows, then, with high probability, all the iterations satisfy the weakly non-expansiveness together-thir leads to the convergence of the ADMM-PnP algorithm.
4.3 CONVERGENCE WITHOUT CONVEXITY OF E
The weakly non-expansiveness based convergence analysis holds under fixed step size (i.c., p) of the ADMM-PuP algorithm, which is consistent with practical implementations in many cases. However, the assumption that the l term is ju-strongly convex is only met by some inverse problems, e.g..
signal denoising and deblurring, but not met by others such as signal compression/recovery and data completion. In this subsection, we remove the convexity assumption and analyze the AC-DC denoiser's properties under the adaptive p-scheme following that in (Chan et al. 2016).
Theorem 3. Suppose that Assumptions
hold. Let D diam suPa,yex || U1l2
Do and define L MD + S. Let Dats ((k)) + zow denote the AC-DC denoiser. Also, assume that the DC step reaches the stationary distribution for each A Then, the following hold:
00, S := infwex V log ( <
Note that Theorems 2 and use this stationary distribution assumption for notation conciseness. For their counterparts removing this assumption, see Appendix F.

---

Published as a conference paper at ICLR 2026
(a) (Boundedness) With probability at least 1 ut, the denoiser Dote) 3 is bounded at each iteration
k i.e. (D (a))2 S c whenever a + (0(1)) <1/M, where G* = (2=)
40x) + 160 20 log 2/v, + 20 g(k) 20, ( (2) ( LL, and Vk > O.
Let vs In 2= 212 k with 7, E (0, 1). Consequently, the denoiser Docs is bounded for all N.,
bn
with corresponding eg and probability at least T}.
(b) (Convergence) Assume there exists R. < oc such that [ - R. Apply the p-increasing 2016) and schedule (a(k), o such that lim (2) +4// + 42v) =
rule in CChan et al.
-26 0, limg-+o0 1-Mo ,(k)
log us
0, lim ((k) 0, limg-roc g((k) 0 02 + (o(1))
1 1/M, Vk N4 for vk In 2m²
bn + 2 Ks with n E (0, 1]. Then, the solution sequence converges to
fixed point with probability at least 1 17.
The proof is relegated to Appendix E Theorem B shows that, with high probability the denoiser is bounded uniformly across all iterations k. Part (b) further shows that, under the proper scheduling of (a(k), (,))) the AD-DC ADMM- -nPP algorithm converges to a fixed point with high probability.
The condition of D < do implies the data space X' has bounded support, which is natural in practice: for images, pixel intensities typically lie within a bounded range such as [0, 1]. Additionally, the condition S < oc ensures that there exists at least one point in X' where the score norm is finite. This prevents pathological cases where the score diverges everywhere (making the distribution degenerate). Together, these conditions guarantee that the score is "well-behaved".
remark is that all theoretical results in this section focus on fixed-point convergence, which is not the strongest form of convergence guarantees. Establishing stronger convergence results, e.g.. stationary-point convergence, for PnP approaches is considered challenging as the objective function implicit (more specifically, h(-) in (2) implicit). Nonetheless, in recent years, some efforts have been made towards establishing stationary pointt convergence for PnP methods under certain types of denoisers (see, e.g., (Hurault et al. 2022a Wei al., 2025
Xu et al., 2025)): more discussions are
in Sec. 5
5 RELATED WORKS
ADMM-PnP PnP has gained much popularity, due to access to data-driven effective denoisers. It has been used in various applications like image restoration (Chan et al.) (0166), data compression (Yuan et al.,
2022 hyperspectral imaging (Liu et al. 2022), and medical imaging (Ahmad et al. 2020).
Theoretical understanding of PnP algorithms with general "black-box" denoisers remains limited. Unlike classical proximal operators, the implicit regularizer h(-)) in (2) handled by data-driven de- noisers is typically unknown. Hence, many results are therefore restricted to fixed-point convergence; see, e.g., (Ryu et al. 2019 Chan et al. 2016). Nonetheless, in certain cases where the denoisers have interesting structures, stronger convergence results can be established. For example, (Xu et al.) 2025) used classical results from image denoising connecting linear denoisers with quadratic h(.)) to show that when linear denoisers are employed, ADMM-PnP converges to KKT points. Hurrull et al.) (2022a) showed stationary-poiin convergence of PnP methods gradient-type denoisers, leveraging the fact that this type of denoisers can be written as proximal operator of a special function (see the nonconvex counterpart in (Hurault et al. 2022b)); Wei et al. (2025) trained denoisers to satisfy a cocoercive conservativity condition, which also ensures convergence of PnP to stationary points associated with an implicit convex h(-). Nonetheless, these results do not cover diffusion- based denoisers. In this work, we generalize the fixed-point convergence proofs in (Chan et al. 2016 Ryu et al. 2019) to accommodate the diffusion score-based AC-DC denoiser.
Recent advances in score-based based generative modeling have motivated their integration into PnP algorithms. One line of work directly replaces the proximal denoiser with a pre-trained scores (Zhu et al. 2023 Li et al. 2024). Alternatively, others embed the score function an explicit regularizer with task-specific loss (Mardani et al. 2024| Renaud et al. 2024a). Deterministic version PnP have also been considered. For example, Wang et al. (2024), ; Song et al. (2023) use unrolled ODE and consistency model-distilled one-step representation to express the target signal, respectively. These methods are similar to (Bora et al. 2017), but with diffusion-driven parameterization.
7

---

Published as a conference paper at ICLR 2026
Original
Measurement
Ours PSNR
DPS PSMR|SSTM|LPIPS 36] 0.82|0.134
DiffPIR |SST| LPPPS
DDRM PSMR|SSIM|| 29.89| 0.72|88.21
LPIPS
99
91 18.07
28. 43. 7.73 188
Figure 2: Inpainting under random missings.
Original
Measurement
Ours PSNR |SSM LPIPS 23 82 141
DAPS-4K PSNR ISIM 21.42| 0.81 (0.167
DiffPIR PSNRISSIM SSIM LPIPS 17.64] (0.800
DPS PSWR [SSIN] 21. 6.81 (0.188
Figure 3: Inpainting under box missing.
Prior works have emphasized the importance of matching the residual noise to the operating range of the PnP denoiser. D-AMP (Metzler et al. [016]] EEsiooluu & Tanc (2018) achieve this via the Onsager correction, which approximately Gaussianizes the residual under compressive sensing problem structures. |Wi et al. (202T) learns a reinforcement learning-based policy to automatically internal parameters, including denoising strength. Unlike AC-DC that provides a generic correction mechanism for a variety inverse problems, these methods either problem specific or require training additional models. Score-based inverse problem solvers have also attempted to "bring" iterates to noisy data manifolds used during training. The work Chung al. (2022) uses a manifold constraint based on gradient of data- data-fidelity, while He et al.] (202)) uses an ofi-the-shelf pretrained neural network to impose a manifold constraint. On the other hand, Zirvi al. (2025) uses the projection of measurement guidance to low-rank subspace, using SVD the intermediate diffusion state, for similar purposes.
all
The idea of adding noise before evaluating our AC step) has been widely considered (Li et al.) 2024) Graikos et al.) [022] Renaud et al., 2024b] Mardani al., 2024). A variant of this called estimation-correttnn idea proposed in (Karras et al. 2022) is used in (Zhu et al. 2023) for this purpose.
functions during optimization procedures (similar to
6 EXPERIMENTS
Dataset and Evaluation Metrics. For all these tasks, we use two datasets: FFHQ 256 x 256 (Karras al. T02T) and ImageNet 256 256 (Deng et al.) (200)). During testing, we randomly sample 100 images from the validation set of each dataset. All the methods use the pre-trained score model in Chung et al. (2023). We use Peak Signal-to-None
Ratio (PSNR) as pixel- wise similarity metric, and Structural Similarity Index (SSIM) and Learned Perceptual Image Patch Similarity (LPPPS) (Zhang al. 2018) as perceptual similarity metries. We report these metrics averaged over the 100 test images for each method and inverse problem.
Task Description. We consider & ~ NN with on 0.05 for all the tasks. (a) For super- resolution,
cubic interpolation method with kernel size 4 for downsampling the resolution by 4 times. (b) For recovery under Gaussian blurring (Gaussian deblurring). a kernel of size 61 and standard deviation is used. (c) As for recovery under motion blurring (motion deblurring). a kemel of size 61 and standard deviation 0.5 used. (d) In inpainting under box mask (box inpainting). approximately centered mask of size 128 128 sampled in image while maintaining the 32
8

---

Published as a conference paper at ICLR 2026
Original
Measurement
AC only 1-0
AC-DC J-10
AC-DC 1-20
Ours PSNR|SSIN (SSIN LPIPS
DAPS-4K [ARR|SSSM]PPPSS
DPS PSNR| SSSI LLPIPS 24.88] 0. 28.
30.78] 0..14
29.97] 0.81 (8.
Figure 4: Recovery under motion blurring.
Figure 5: Influence of DC steps in the denoiser.
pixel margin in both spatial dimensions of the input image. (e) For inpainting under random missings (random inpainting), 70% of the pixels uniformly sampled be masked, and a scaling of 2 was used in high dynamic range (HDR) before clipping the values. (f) For phase retrieval, similar as in prior works (Wu al. 2024; Mardani et al. 2024), we use oversampling by factor of 2. g) For deblurring under nonlinear blurring, we use the operator in (Tran et al. (021) with default settings.
Baselines. We set of baselines, namely, DPS (Chung al. 2023). DAPS (Zhang al. 2024).
DDRM (Kawar et al., 2022). DiffPIR (Zhu et al. 2023), RED-dif diff (Mardani et al. 2024). DPIR
(Zhang al. 2022). DCDP (Li
al. 2025). PMC
(un et al. 2024).
Hyperparameter Settings. We adopt linear schedule for o with range (0.1, 10] over W decay window i.e. (*) max 10.x 10 (10 0.1) k/W). The maximal number of iterations for proposed method is set to = W + 10. At iteration k; we use J = 10 DC steps, and the schedules (14) = (0) and 0 (*) = 0.1/Vots, We use gradient descent with Adam optimizer (Kingma & Ba 2015) for solving each regularized maximum likelihood subproblem (7)). This subproblem is optimized for maximum of 1000 iterations with convergence detected when the loss value increases more than A 1 10-1 consecutively for 3 iterations window. We conduct our experiment with two variants based on the third stage: using Tweedie's 's lemma (denoted as "Ours-tweedie") and
10-step ODE based denoiser (Zhang al. 2024 Karras al. 2022) (denoted as "Ours-ode"). We
use the preconditioning in Karras et al. 2022) while using the pretrained diffusion models.
Qualitative Performance. Figs 2 and show reconstructions under inpainting under random missings, inpainting under box missing, and motion deblurring. It can be seen that our method is able to recover the image that is comparatively natural looking with less noise and artifacts, while being consistent with the measurements. On the other hand, images recovered with DiffPIR appears to suffer from noise and artifacts, whereas DPS leads to meaaureeeent-icconiisse
reconstructions.
Our method outperforms others while other methods appear to either be blurred or contain noisy artifacts in the recovered image. Recovery by DPS is less consistent with the original image; the pattern on the child's clothing is completely lost.
Quantitative Performance. Table summarizes PSNR, SSIM and LPIPS averaged over 100 images on FFHQ and Imagenet datasets. In almost all of the inverse problems, both of our variants (Ours- tweedie and Ours-ode) achieve the best or second-ber performance in terms of all metrics. Our method significantly outperforms other PnP baseline methods considered, namely, DDRM, DifIPIR and RED-diff. This demonstrates the effectiveness of our AC-DC denoiser.
Effectiveness of DC. To perform ablation study on the DC stage, consider the challenging phase retrieval problem. Fig. shows the output of ADMM-PaP with our AC-DC denoiser with different numbers of DC iterations J. With J = (disabling DC step), artifacts remain severe. Increasing J progressively results in cleaner images.
More Details and Additional Experiments. More details and experiments in appendices.

---

Published as a conference paper at ICLR 2026
Table I: Reconstruction metries (100 images) on FFHQ / ImageNet. Bold: best, blue: 2nd best.
FRIQ
ImageNet
FFII
ImugeNei
Tisic
Method
PSNRT
SSIMT
LPIPS.
SSIM)
LPIPS]
Task
Methed
PSNRY
SSIM
LPIPS
PSNKT
SSIMT
L
Supercellution
tee--eeedde
30,439
0.857
0.178
27.318
0.717
01220
Ours coe--c
30,402
0.853
0.175
27.199
0.705
0.281
Ous--ode
29.991
0.845
0.156
26.919
0.700
0.276
Gousiaa
Ours ode
30.019
0.841
0.158
26,899
0.690
0.282
(80)
DPS
DAPS
24.828
29.529
0.705
0.814
0.167
0.257
22.785
26.653
0.680
0.549
(1411
0.366
DPS
DAPS
26. 19k
20.75 797
0.813
0,750
0.207
0.157
26,886
23.995
0.575
D.678
0.328
0.260
DDKM
27.145
0.782
0.261
105
0.683
(1206
DMTPIR
25. 25.148
0.699
0.230
22.756
0.508
0.314
DMAPER
26.771
0.749
0.208
23.558 554
0.541
0.336
DPIR
28. 87.
0.833
0.228
76
0.700
0.314
RED-JiT
16.831
0.423
0.547
66.66
0.309
0.519
TXCD
16. 16.821
0.171
0.642
155 102
0.136
0.620
DPIR
28,819
0.826
0.254
26.50 524
0.699
0.334
PMC
20.172
0.638
0.344
24.103
0.515
0.115
DCDP
27.761
0.639
0.332
24.517
0.525
0.361
PMC
25.774
22.534
0.456
twee-t
24.025
0.859
0.131
21.626
0.789
0.222
Our tweedie
32,844
0.906
0.122
29.564
0.517
0.154
Inpeniing
[Box)
DAPS
Ours-ode
23.342
23.643
0.815
0.837
0.136
0.146
21.305
20.618
0.743
0.774
0.227
0.199
Impaining
(Kandom)
Ouss ode
DAPS
31,652
32.127
0.847
0.854
0.124
0
28.137
28.733
0.751
0.795
0.162
0.148
DERPIR
DPS
20.934
23:488
0.56]
0.817
0.291
0.164
19.565
19:933
0.562
0.677
0,309
0.3:2
DPS
29.084
0.828
0.181
26,099
0.678
(1318
RED-dit
18.715
0.523
0.364
18.075
03.99
0.371
DDKM
28.909
0.847
0.178
27.833
0.778
01202
DCDP
25,250
0.754
0.163
20.991
0.727
0.195
DIPPER
28.558
0.709
0.230
26.93 913
0.679
0.223
PMC
14.828
0.697
0,318
15,550
0.666
0.326
RED-dif
20.561
0.630
0.275
20.948 948
0.464
01315
PMC
23.289
0.755
0.263
25.965 965
0.636
0.34
Reneval
Ours-cwaddic
27,944
0,793
0.201
17:770
0.440
0.471
Plass
Ours-ode
27.095
0.257
0.257
16:015
0.339
0.539
Modion
Debuur
Ours-bweedie
Oure-ode
29.648
30,033
0:841
0.854
0.179
0.154
27.149
26.615
0.717
0.604
0.280
0.275
DPS
DAPS
26.707
11.627
0.366
0.749
0.658
0.230
16.445
9.434
0.395
0.216
0.768
0.512
DAPS
29/051
0.815
0.175
26.571
0.680
0.376
RET-SI
15411
(1-90)
0.490
12,852
0.204
0.695
DPS
23:257
0.663
0.265
19,613
0.451
0.451
DCDP
20.026
0.540
0.124
12.257
0,212
0.665
PMC
19:450
0.590
0.426
21,608
0.480
0.510
PMC
10:421
0.267
0,783
8.636
0.129
0,890
CONCLUSION
We introduced the AC-DC denoiser, a score-based denoiser designed for integration within the ADMM-PnP framework. The denoiser adopts a three-stage structure aimed at mitigating the mismatch between ADMM iterates and the noisy manifolds on which score functions are trained. We established convergence guarantees for ADMM-PnP with the AC-DC denoiser under both fixed and adaptive step size schedules. Empirical results across a range of inverse problems demonstrate that the proposed method consistently improves solution quality over existing baselines.
Limitations. While our analysis provides initial insights, several aspects merit deeper understanding. The second convergence result relaxes convexity by allowing adaptive step sizes, though such schedules are arguably less appealing in practice. Our experiments, however, suggest that constant step sizes also perform well for nonconvex objectives; is therefore desirable to establish convergence guarantees for constant step sizes in such settings. In addition, our result ensures the stability of the ADMM method, but does not directly explain the reason why the AC-DC denoiser attains high-quality recovery; recoverability and estimation error analyses are also desirable. On the implementation side, the noise schedules used in the AC and DC stages are currently guided by empirical heuristics. Designing problem-adaptive scheduling strategies may further improve both convergence speed and robustness. Additionally, each iteration of AC-DC denoiser needs multiple score evaluations. Reducing the required NFEs could significantly improve its efficiency.
10

---

Published as a conference paper at ICLR 2026
This work focuses exclusively the theory and methodology of solving inverse problems. It does not involve human subjects, personal data, or any sensitive procedures.
Ethics Statement:
Reproducibility Statement: The source code is provided part of the supplementary material. All assumptions, derivations and necessary details regarding the theory and experiments are included in the appendices.
Acknowledgment: This work was supported in part by the National Science Foundation (NSF) under Project NSF CCF-2210004.
REFERENCES
Rizwan Ahmad, Charles A. Bouman, Gregery T. Buzzard, Stanley Chan, Sizhuo Liu, Edward T. Rechorst, and Philip Schniter. Plug-and-play methods for magnetic resonance imaging: Using denoisers for image recovery. IEEE Signal Processing Magazine, 37(1):05--116 January 2020.
ISSN 1558-0792. doi: 10.1 1099mmpp.219..24444700 URL Btllhttp:// /xx http:/xxoo.o..ooooooooooo wttp://doo.oog....o.ooo// MSP 201 024992294470
Ismail Alkhouri, Shijun Liang, Rongrong Wang, Qing Qu, and Saiprasad Ravishankar. Robust
physics-based deep mri reconstruction via diffusion purification, 2023. URL httthhtss://araa
org/ aa/aabbb/230.. 05794
Ismail Alkhouri, Shijun Liang, Evan Bell, Qing Qu, Rongrong Wang, and Saiprasad Ravishankar. Image reconstruction via autoencoding sequential deep image prior. In The Thirty-eighth Annual Conference on Neural Information Processing Systems. 2024. URL /
net
Simon R Arridge. Optical tomography in medical imaging. Inverse problems, 15(2):441, 1999.
Heinz H Bauschke, Patrick L Combettes, Heinz H Bauschke, and Patrick L Combettes. Comex analysis and monotone operator theory Hilbert spaces. Springer, 2017.
Andrew Bennett. Inverse methods in physical oceanography. Cambridge university press, 1992.
Ashish Bora, Ajil Jalal, Eric Price, and Alexandros Dimakis. Compressed sensing using generative models. In International conference on machine learning, pp. 537-546. PMLR, 2017.
Stephen P Boyd and Lieven Vandenberghe. Convex optimization. Cambridge university press, 2004.
Herm Jan Brascamp and Elliott H Lieb. On extensions of the brunn-minkowski and prékopa- leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. Journal of Functional Analysis, 22(4):366-38)) 1976. ISSN 0022- 1236. doi: https::/dooi.or//1...
wtt../////.0100100222220100 1016/0022 1236(76)90004-5.5 5. URL https: www. sci sww.ssceineccreetttttttt....ee..
com/ sci
www.arttccclllpppp.mmmm http:/a1......1112/22/0// 0111222677660000 6900045
Felix E. Browder. Nonexpansive nonlinear operators in banach space. Proceedings of the National Acauemy of Sciences of the United States of America, 54(4): 1041-1044, 1965. ISSN 00278424,
10916490. URL URL www.
Gregery T Buzzard, Stanley H Chan, Suhas Sreehari, and Charles A Bouman. Plug-and-ple unplugged: Optimization-free reconstruction using consensus equilibrium. SIAM Journal on Imaging Sciences, 1 11(3):2001-2200, 2018.
Stanley Chan, Xiran Wang, and Omar Elgendy. Plug-and pld--and admm for image restoration: Fixed point convergence and applications. IEEE Transactions on Computational Imaging, PP, 05 2016. doi: 10.1109/TC1.201..22223
Stanley H. Chan. Performance analysis of plug-and sdug-and--lay admm: A graph signal processing per- spective. LuXXvv1189.0002200 May 2019. doi: 10.48550/arxiv. 0000000020. URL http:
//arxiv. //aaxiv.err/abs/1B8008.
00020
arXiv: 809.00020 [eess].
Xi Chen, Simon S. Du, and Xin T. Tong. On stationary point hitting time and ergodicity of stochastic gradient langevin dynamics. Journal of Machine Learning Research, 21(68): 2020. URL http:/ /ttp:///mm wwwggpapprs//vi/e
327 html
11

---

Published as a conference paper at ICLR 2026
Hyungjin Chung, Byeongsu Sim, Dohoon Ryu, and Jong Chul Ye. Improving diffusion models for inverse problems using manifold constraints. Advances in Neural Information Processing Systems, 15:25683-2569229 2022.
Hyungjin Chung, Jeongsol Kim, Michael Thompson Mecann, Marc Louis Klasky, and Jong Chul Ye. Diffusion posterior sampling for general noisy inverse problems. In International Confer- ence on Learning Representations, 2023. URL
OnD9ZGAGTOk!
B Combal, Frédéric Baret, M Weiss, Alain Trubuil, D Macé, A Pragnere, R Myneni, Y Knyazikhin, and L Wang. Retrieval of canopy biophysical variables from bidirectional reflectance: Using prior information to solve the ill- ill-posed inverse problem. Remote sensing of environment, 84(1):1-15, 2003.
Patrick L. Combettes and Isao Yamada. Compositions and convex combinations of averaged nonexpansive operators. Journal of Mathematical Analysis and Applications, 425(():55-70) 70, 2015. ISSN 0022-247X. doi: http://dbbborr//10.01....mmaaaaa
1.444. URL https
/ /www www..cceeeeee...cco....... com htt.a//t.................///////
Kostadin Dabov, Alessandro Foi, Vladimir Katkovnik, and Karen Egiazarian. Image denoising by sparse 3-d transform-domann collaborative filtering. IEEE Transactions Image Processing, 16 (8):2080-2095, 2007. doi: 10. 10.9000....2000.99012222
Arnak S. Dalalyan and Avetik G. Karagulyan. User-friendly guarantees for the langevin monte carlo with inaccurate gradient. Stochastic Processes and their Applications. December 2019. ISSN 03044149, doi: 10.
02..00..16..
[math].
Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hier- archical image database. 2009 IEEE Conference on Computer Vision and Puttern Recognition, pp. 248-255, 2009. doi: 10.1 100/CVPR.20099 2000066888..
Alex Dytso, H. Vincent Poor, and Shlomo Shamai. A general derivative identity for the conditional mean estimator in gaussian noise and some applications, 2021. URL htttss:///aai::t. org/
abs/2104, 01883
Jonathan Eckstein and Dimitri P Bertsekas. On the douglas rachford splitting method and the proximal point algorithm for maximal monotone operators. Mathematical programming, 55: 293-318, 1992.
Ender M. Eksioglu and A. Korhan Tane. Denoising amp for mri reconstruction: Bm3d- mmi. SIAM Journal on Imaging Sciences, 11(3))2000-2109, January 2018. doi: 10.1137/18M1116655
Michael Elad and Michal Aharon. Image denoising via sparse and redundant representations learned dictionaries. IEEE Transactions on Image Processing, 15(12):3736-3745 2006. doi:
10.1100//TP22000.8881999 1966..
Dara Entekhabi, Hajime Nakamura, and Eni G Njoku. Solving the inverse problem for soil moisture and temperature profiles by sequential assimilation multifrequency remotely sensed observations. IEEE Transactions Geoscience and Remote Sensing, 32(2):438 1994.
Pontus Giselsson. Tight global linear convergence rate bounds for douglas-rachfone splitting.
Journal of Fixed Point Theory and Applications, 19:2241 2270, 2015. URL httpps:/appi
semant iesaantisschllast www.aar.....oooooooooooooao
62917
Alexandros Graikos, Nikolay Malkin, Nebojsa Jojic, and Dimitris Samaras. Diffusion models as plug- and-play priors. In Alice H. Oh, Alekh Agarwal, Danielle Belgrave, and Kyunghyun Cho (eds.), Advances Neural Information Processing Systems, 2022. URL https: htttps://ppeereiieeet
C. Hatsell and L. Nolte. Some geometric properties of the likelihood ratio (corresp.). IEEE
Transactions Information Theory, 1765)6616-618, 1971. doi: 10. 10.11090 199./T..197.1115.41
12

---

Published as a conference paper at ICLR 2026
Yutong He, Naoki Murata, Chich- Lai, Yuhta Takida, Toshimitsu Uesaka, Dongjun Kim, Wei- Hsiang Liao, Yuki Mitsufuji. J Zico Kolter, Ruslan Salakhutdinov, and Stefano Ermon. Manifold preserving guided diffusion. In The Twelfth International Conference on Learning Representations, 2024. URL wttps: http///ppeeevvveeeeeee/ / htt.://wwwweeooooooooooooooo.oa.
net/ forum? dd=o:220000000.ommmmmm
Wenrui Hu, Dacheng Tao. Wensheng Zhang.
Yuan Xie, and Yehui Yang. The twist tensor nuclear norm for video completion. IEEE Transactions on Neural Networks and Learning Systems, 28(12):
2961-2973, 2017. 2017. doi: 10.1099TNNNLS2211:2261111
Samuel Hurault, Arthur Leclaire, and Nicolas Papadakis. Gradient step denoiser for convergent plug-and-play. (arXiv:2110.03220) February 2022a. doi: 00.4850//XXxv.211000322 URL
03220 arXiv:21 10.03220 [es].
Samuel Hurault, Arthur Leclaire and Nicolas Papadakis. Proximal denoiser for convergent plug-and-play optimization with nonconvex regularization. In Proceedings of the 39th Inter- national Conference on Machine Learning, pp. 9483-9505. PMLR, 2022b. URL. https
//proceedings ml www..r.reee/i////urrraae
html
Kyong Hwan Jin, Michael T McCann, Emmanuel Froustey, and Michael Unser. Deep convolutional neural network for inverse problems in imaging, IEEE transactions on image processing. 26(9): 4509-4522, 2017.
Tero Karras, Samuli Laine, and Timo Aila. A style-based generator architecture for generative adversarial networks. IEFE Trans. Pattern Anal. Mach. Intell., 43(12):4217-4222, December 2021. ISSN 0162-8828. doi: 10.1109/PAMM22200..27000
htttps:/ URL/atpss//ooo.oor/20...11
TPAMI 2020 9970919
Tero Karras, Miika Aittala, Samuli Laine, and Timo Aila. Elucidating the design space of diffusion- based generative models. In Proceedings of the 36th International Conference on Neural Infor- mation Processing Systems, NIPS '22, Red Hook, NY, USA, 2022. Curran Associates Inc. ISBN
978171387100 1088.
Bahjat Kawar, Gregory Vaksman, and Michael Elad. SNIPS: Solving noisy inverse problems stochastically. In A. Beygelzimer, Y. Dauphin, P. Liang, and Wortman Vaughan (eds.), Advances in Neural Information Processing Systems, 2021. URL 1d=pBXOx dxYAN
net netfforumt forum?
Bahjat Kawar, Michael Elad, Stefano Ermon, and Jiaming Song. Denoising diffusion restoration models. In Alice H. Oh, Alekh Agarwal, Danielle Belgrave, and Kyunghyun Cho (eds.), Advances in Neural Information Processing Systems, 2022. URL httpthtts///o/o http:::::///ppeeevveeeee.e net few.orooiiwweeeeenn forum? Id=kxXvopt
Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In Yoshua Bengio and Yann LeCun (eds.), 3rd International Conference on Learning Representations, ICLR 2015, San Diego CA, USA, May 7-9, 2015, Conference Track Proceedings, 2015. URL http:
7 //arxiv... org/ abs aaaaabas/1112 111//1111112 6980
B. Laurent and P. Massart. Adaptive estimation of quadratic functional by model selection. The Annals of Statistics, 28(5): October 2000. ISSN 0090-5364, 2168-8966. doi: 10.121 20.1214/aos/101 5957395.
Xiang Li, Soo Min Kwon, Ismail R. Alkhouri, Saiprasad Ravishankar, and Qing Qu. Decoupled data consistency with diffusion purification for image restoration. (2XXiv2400306054)) May 2024. doi: 10.48550/arxx..2403.006 arXiv:2403.00054 [eess].
URL http://aaxxv..rr//abbb//
Xiang Li, Soo Min Kwon, Shijun Liang, Ismail R. Alkhouri, Saiprasad Ravishankar, and Qing Qu Decoupled data consistency with diffusion purification for image restoration, 2025. URL atttps://arxiss www./orggg/ooo.oo abs /2003. 06054
Kewei Liang. Homocentric convergence ball of the secant method. Applied Mathematics-A Journal of Chinese Universities, 22:353-365, 2007.
13

---

Published as a conference paper at ICLR 2026
Yun- Yang Liu, Xi-Le Zhao, Yu-Bang Zheng, Tian-Hui Ma, and Hongyan Zhang. Hyperspectral image restoration by tensor fibered rank constrained optimization and plug-and aad--addppyyy regularization. IEEE Transactions on Geoscience and Remote Sensing, 60:1-17, 2022. doi: 10.1109/TGRS.2000. 3045169.
Morteza Mardani, Jiaming Song, Jan Kautz, and Arash Vahdat. A variational perspective on solving inverse problems with diffusion models. The Twelfth International Conference on Learning Representations, 2024. URL http:::
net httpr//wwweeeeeeooooooeee-aaa---
J. C. Mattingly, A. M. Stuart, and D. J. Higham. Ergodicity for sdes and approximations: locally lipschitz vector fields and degenerate noise. Stochastic Processes and their Applications, 101(2): 185-232, October 2002. ISSN 0304-4149. doi:
Chenlin Meng, Yutong He, Yang Song, Jiaming Song, Jiajun Wu, Jun-Yan Zhu, and Stefano Ermon. SDEdit: Guided image synthesis and editing with stochastic differential equations. In International Conference on Learning Representations, 2022. URL httt:p:s//peereeeee..etttt
id=aBsCJcPu t
Christopher A. Metzler, Arian Maleki, and Richard G. Baraniuk. From denoising to compressed sensing. (arXiv: 406.4175), April 2016. doi: org/abs/
110.000aaXxx..44.4117.. URL http:://axaxt
abs 1406.4466.0 4175
#rXiv:1406.4177. [cs].
Weili Nie, Brandon Guo, Yujia Huang, Chaowei Xiao, Arash Vahdat, and Animashree Anandkumar. Diffusion models for adversarial purification. In Kamalika Chaudhuri, Stefanie Jegelka, Le Song, Csaba Szepesvari, Gang Niu, and Sivan Sabato (eds.), Proceedings of the 39th International Conference on Machine Learning. volume 162 of Proceedings of Machine Learning Research, pp. 16805-16827. PMLR, 17-23 Jul 2022. URL https: httpt:ttttpp//eeeeeeeeo http:///ccccceeeeeeeeeeee .mrr. www..prees//ee Iiie2aa html.
D.P. Palomar and S. Verdu. Gradient of mutual information in linear vector gaussian channels. IEEE Transactions on Information Theory, 52(1):141-154, 2006. doi: 10.1109/TT.22005884444,
Leandro Pardo. Statistical inference based on divergence measures. Chapman and Hall/CRC, 2018.
Xinyu Peng, Ziyang Zheng, Wenrui Dai, Nuoqian Xiao, Chenglin Li, Junni Zou, and Hongkai Xiong, Improving diffusion models for inverse problems using optimal posterior covariance, 2024. URL
/ 2402 24402 02149
https::/arxiv> wwg. awtpg//a/////
Maziar Raissi, Paris Perdikaris, and George F. Karniadakis. Physics-infomme neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics, 378:686-707, 2019.
Hongmin Ren and loannis K. Argyros. On the complexity of extending the convergence ball of wang's method for finding a zero of a derivative. Journal of Complexity, 64:101526, 2021. ISSN 0885- 064X. doi: https///doooog///0.0100fjoooo/o
URLhttpa www. scw.ncedreeetcttttteeecc..eeee..
com/ http://ccaccccrrtaeeeeee//////a0
Hongmin Ren and Qingbiao Wu. Convergence ball and error analysis of family of iterative
methods with cubic convergence. Applied Mathematies and Computation, 209(2):369-378 2009. ISSN 0096-3003, doi: org.apccc....c.ooo......oooooooa
12500. URL https: /www
sciencedirect. we..enceeirecttccom/ei.eeieeeeee
Marien Renaud, Jean Prost, Arthur Leclaire, and Nicolas Papadakis. Plug-and-play image restoration with stochastic denoising regularization. In Proceedings of the 41st International Conference
Machine Learning, (CML'24. JMLR.org 2024a.
Marien Renaud, Jean Prost, Arthur Leclaire, and Nicolas Papadakis. Plug-and-play image restoration with stochastic denoising regularization. CoRR, abs/2402.01779, 2024b. URL. httpt:ht////t //dii
org a hTr//10.4850/arx22 2402. 01779
Herbert Robbins. An empirical bayes approach to statisties. Proceedings the Third Berkeley Sympo- sium on Mathematical and Statistical Probability, 1, 01 1992. doi: 1-111/798-----1---99---
26.
14

---

Published as a conference paper at ICLR 2026
Ernest Ryu, Jialin Liu, Sicheng Wang, Xiaohan Chen, Zhangyang Wang, and Wotao Yin. Plug- and-play methods provably converge with properly trained denoisers. Kamalika Chaudhuri and Ruslan Salakhutdinov (eds.), Proceedings of the 36th International Conference Learning, volume 97 of Proceedings of Machine Learning Research, pp. 35547. PMLR, 09-15 Jun 2019.
Machine
pttctps::/pprceedingg ngs m 1 wwwpppsssssspppp ruu wttp.nnn.........//.1..1... 9a .html
Oguz Semerci, Ning Hao, Misha E. Kilmer, and Eric L. Miller. Tensor-based formulation and nuclear norm regularization for multienergy computed tomography. IEEF Transactions on Image Processing, 2374):1678-1693 2014. doi: 10.110991122211233555544
Viraj Shah and Chinmay Hegde. Solving linear inverse problems using gan priors: An algorithm with provable guarantees. In 2018 IEEE international conference acoustics, speech and signal processing (ICASSP), pp. 4609-4613, IEEE, 2018.
Bowen Song, Soo Min Kwon, Zecheng Zhang, Xinyu Hu, Qing Qu, and Liyue Shen. Solving inverse problems with latent diffusion models via hard data consistency. October 2023. URL
https://epenzrviier net fortm/ormm?eem
Jiaming Song. Arash Vahdat, Morteza Mardani, and Jan Kautz. Preudoinverss--guided diffusion models for inverse problems. September 2022a. URL pettpss///peerrvvie....
1d=9
gaMAEMRKQ
Yang Song, Jascha Sohl-Dickstein, Diederik Kingma, Abhishek Kumar, Stefano Ermon, and Ben
Poole. Score-based generative modeling through stochastic differential equations. In 9th Interna- tional Conference on Learning Representattinss ICLR 2021, Virtual Event, Austria, May 3-7, 2021.
OvenReview.net, 2021. URL httpittt///ppeeeeiieee http://wwe.eee.eeeeeeeeeeeee http:///wwwweeeeeee//oor/aa/aaaa
Yang Song, Liyue Shen, Lei Xing, and Stefano Ermon. Solving inverse problems in medical imaging with score-based generative models. In International Conference on Learning Representations, 2022b. URL URLhttpp::/cppereviiee wttp://ww.ee//oueueeeeee httrr//ormmrirreeee 2ott:/Eoorum2ddddaaeRCH1
net/
Yu Sun, Brendt Wohlberg, and Ulugbek S. Kamilov. An online plug-and and-play algorithm for reg- ularized image reconstruction. IEEE Transactions Computational Imaging, 5(3):395 September 2019. ISSN 2333-9403, 2334-0118, 2573-0436. doi: 10.1109/TCL22011288988
arXiv:1809 809.00999.. [es].
Yu Sun, Zihui Wu, Yifan Chen, Berthy T. Feng, and Katherine L. Bouman. Provable probabllitti imaging using score-based generative priors, 2024. /BLL:ttts.//axxx..rrr//////
10835
Albert Tarantola. Inverse problem theory and methods for model parameter estimation. SIAM, 2005.
Afonso M. Teodoro, José M. Bioucas-Dias, and Mário A. T. Figueiredo. Scenc-adapted plug-and-play algorithm with convergence guarantees. (arXiv: 172.00445) November 2017. doi: 10.48550/ 170.0045...
wwtp:/iwwwrr/aaa/////aaaa 02445
02445 arXiv:1702.02445 [cs].
Edward Charles Titchmarsh and David Rodney Heath-Brown. The theory of the Riemann zeta- function. Oxford university press, 1986.
Phong Tran, A. Tran, Quynh Phung, and Minh Hoai. Explore image deblurring via encoded blur kernel space. 202/ IEEE/CVE Conference on Computer Vision and Pattern Recogni- tion (CVPR), pp. 11951-11960, 2021. URL httpps://pp http://pp....aaa..mmmmmm /
www.enticschol............. org/
Corpus1D:225328533
G. E. Uhlenbeck and I.. S. Omstein. On the theory of the brownian motion. Phys. Rev, 06:823-841, Sep 1930. doi: 10. 10.110%/hhyseev.36.333 URL wtttth:t///iinte wwt.:aa.pp.....ooo.o..// wotg//o////.oo111o
PhysRev. 36. 823
Dmitry Ulyanov, Andrea Vedaldi, and Victor Lempitsky. Deep image prior. International Jour- nal of Computer Vision, 128(7):1867-1888, March 2020. ISSN 1573-1405. doi: 10.1007/ 020-01303-
sl 1223-002-001303- URL dww...oo...ooooo http:/111.10001112111
Cédric Villani al. Optimal transport: old and new, volume 338. Springer, 2008.
15

---

Published as a conference paper at ICLR 2026
Martin J Wainwright. High-dimensional statistics: A non-asymptotic viewpoint, volume 48, Cam- bridge university press, 2019.
Hengkang Wang. Xu Zhang, Taihui Li, Yuxiang Wan, Tiancong Chen, and Ju Sun. DMPlug: A plug-in method for solving inverse problems with diffusion models. In The Thirty-eighth Annual Conference on Neural Information Processing Systems, 2024, URL
Sttt//orum..1d=111820
Yinhuai Wang. Jiwen Yu, and Jian Zhang. Zero-shot image restoration using denoising diffusion null-space model. In The Eleventh International Conference on Learning Representations, 2023. wwnee/foorie forum id mieiiiieieM
URL /RRLhttts////peeeeeeeee
net
Deliang Wei, Peng Chen, Haobo Xu, Jiale Yao, Fang Li, and Tieyong Zeng. Learning cocoercive con- servative denoisers via helmholtz decomposition for poisson inverse problems. (arXiv:2505.08909) October 2025. doi: 10.485500arxiv.222 www...i......00090. URL http://aaxi...cr//aa////// 08909 arXiv:2505.086 0009009000999111 [cs].
Kaixuan Wei, Angelica Aviles-Riveno Jingwei Liang, Ying Fu, Hua Huang, and Carola-Bibiane Schönlieb. Tfpnp: Tuning-free plug-and-play proximal algorithm with applications to inverse imaging problems. (arXiv:0012.05700) 2021. doi: 14..50///2X...22222057.... URL h
arXiv:2012.05703 [cs].
Zihui Wu, Yu Sun, Yifan Chen, Bingliang Zhang, Yisong Yue, and Katherine Bouman. Principled probabilistic imaging using diffusion models as plug-and-play priors. In The Thirty-eighth Annual Conference on Neural Information Processing Systems, 2024. URL httpt::t///ee/ httppsss//ppereeeiiieee
net/forum? io- (at.ooruumnixxx:000ET7
Zhisheng Xiao, Karsten Kreis, and Arash Vahdat. Tackling the generative learning trilemma with
denoising diffusion GANs. In International Conference on Learning Representations, 2022. URL
net/ wttf//oormmddppppp httts://openreviice htgp:/rrereeeenttteoooooorto---- (o
Le Xu, Lei Cheng, Junting Chen, Wenqiang Pu, and Xiao Fu. Radio map estimation via latent domain plug-and-play denoising, 2025. URL http:: /ttp///rvvvvvv/// wwtp:/wwwwww...//........./ 13472
Jianchao Yang, John Wright, Thomas S. Huang, and Yi Ma. Image super-resolution via sparse representation. IEEE Transactions on Image Processing, 19(11):2811-2873, 2010. doi: 10.1109/ TIP.2010.2050625.
Yonghong Yao, Haiyun Zhou, and Yeong-Cheny Liou. Iterative algorithms with variable anchors for non-expansive mappings. Journal of Applied Mathematics and Computing, 28:39-49, 2008
Xin Yuan, Yang Liu, Jinli Suo, Frédo Durand, and Qionghai Dai. Plug-and-play algorithms for video snapshot compressive imaging. IEEE Transactions on Pattern Analysis and Machine Intelligence, 44(10):093-7111, 2022. doi: 2..1109/TAAM.2222..00000
Bingliang Zhang, Wenda Chu, Julius Berner, Chenlin Meng, Anima Anandkumar, and Yang Song. Im- proving diffusion inverse problem solving with decoupled noise annealing. CORR, abs/2407.01521,, 2024. URL/httss://ddat http////oo.oo/.///////////////// 48550/ arw.v...vvvv 2407 01521
Kai Zhang, Yawei Li, Wangmeng Zuo, Lei Zhang. Luc Van Gool, and Radu Timofte. Plug-and-play image restoration with deep denoiser prior. IEEE Transactions on Pattern Analysis and Machine doi: 10. 1011009AM..2221.300000
Intelligence, 44 (10))63606-6636 6376, 2022.
Richard Zhang, Phillip Isola, Alexei A. Efros, Eli Shechtman, and Oliver Wang. The Unrea- sonable Effectiveness of Deep Features as Perceptual Metric In 2018 IEEE/CVE Con- ference on Computer Vision and Pattern Recognition (CVPR), pp. 586-595, Los Alamitos, URL
CA. USA, June 2018 IEEE Computer Society. doi: 10. 10.11000/VPR..018... https://dos eesecooptteesoccceee...... wtw.:iiitttorrogr.....o 100.1199/CVR. 2018.00068
Yuanzhi Zhu, Kai Zhang, Jingyun Liang, Jiezhang Cao, Bihan Wen, Radu Timofte, and Luc Van Gool. Denoising diffusion models for plug- sigg-addplaa image restoration. In /EEE/CVF Conference Computer Vision and Pattern Recognition, CVPR 2023 Workshops, Vancouver, BC, Canada, June 17-24, 2023, pp. 1229. IEEE, 2023, doi: 10. 10..1099CCVW.5992200000.000 https: / org. 10. 1009 CVP RW5 9228. 00129
URL
16

---

Published as a conference paper at ICLR 2026
Rayhan Zirvi, Bahareh Tolooshams, and Anima Anandkumar. Diffusson state- saae-guided projected gradi- ent for inverse problems. In The Thirteenth International Conference on Learning Representations 2025. URL bttpps:: :
17

---

Published as a conference paper at ICLR 2026
A PRELIMINARIES
A.1 NOTATION
Table 2: Summary of notation
Symbol
Description
Rd
The unknown signal image be recovered Measurements, with S d
R
ith element of measurement y Measurement operator Support of æ Support of æt
A IRd + R"
X'
E
Additive measurement noise
ax
noisy data by using forward diffusion process with noise or noisy data by using forward diffusion process with noise (k) data-fidelity loss (e.g. ly A(=))
A(y))A(x)))
h(x)
structural regularization prior (enforced via denoiser)
P >
ADMM penalty parameter
Prox(.)
Proximal operator
(k)
Scaled dual variable in iteration of ADMM
(R)
Auxiliary variable iteration & of ADMM Pre-denoising input to the PnP denoiser Noise level schedule for the AC-DC denoiser Variance parameter in the AC-DC prior for directional correction.
g(k) (k+1) (1)
(A)
(((
12 ~ ~N000
Multivariate standard gaussian random variable Pretrained score function
s0(x, 0) R Va p(x + on)
K
Maximum iteration of ADMM
Total iteration of directional correction at for each denoising Smoothness constant of V log Pdata (Assumption 2
M
M,
Smoothness constant of V log Pt
T>0
Maximum time steps used for diffusion
Mo(t)
Manifold of xt
Ma
Manifold of æ, where / E [0, 7] such that a(1) = (k)
(z(N(
AC-DC denoiser at kth iteration
(amn(Z) = (2)-2
Residual of AC-DC denoiser at kthh iteration
I(x) = x
Identity mapping function Identity matrix vector of values 0
Ti T2(2)= (T(((()))
Concatenation of functions Tj and 2-norm of vector
|0l||
Cov(.)
Covariance matrix
A.2 DEFINITIONS
u-strongly convex function (Boyd & Vandenberghe 2004). A differenniable function f 1 u-strongly convex for certain > Off
f(y) 2 f(x) + Vf(x) - x)) + "
1 2=12
(17)
for all x, y R
The notion of of nonexpansive and averaged nonexpansive have been widely used in the convergence
analysis of various nonlinear problems (Combettes & Yamada 2015 Yao et al. 2008 Eckstein &
18

---

Published as a conference paper at ICLR 2026
Bertsekas 1992).
We use the generalized form of both nonexpansive and averaged nonexpansive operator for establishing the ball convergence in our method.
Nonexpansive function (Browder) 1965). A function T IRd IRd is nonexpansive if T is nonexpansive function if there exists e E [0, 1] such that
(T(x) - T ( - 1|
y112
(18)
for all x, E IR4.
0-averaged fuunction (Combettes & Yamada 2015). A mapping T' Rd Rd is defined be 0 0-averaged for constant 0 E (0, 1) there exists nonexpansive operator R R - R such that I [ + PR.
The notion of relaxed bound T& 2
yll 8(h), Vx, y E X' was used study and show the convergence in Yao al. (2008) when Ck=1 |8(k) oc. We define similar weaker form of nonexpansive function and Hvvrrgee functions below.
(gy
S 2
8- 5-weakly nonexpansive function. mapping T IRd - Rd is said be 5-weakly nonexpansive for 5 2 0 if there exists E [0, such that
[T(x) 11 2 212 v12+82 +
(19)
for all y I
6-weakly d-averaged function. A mapping T : Fd > Rd is defined to be 5-weakly 9-averaged function for certain 5 2 0 and 0 E (0, if there exists 6-weakly nonexpansive function R Rd > Rd such that = 0R + (1 - O)T.
Sub-Gaussian random vector. A random vector æ E Rd (with mean E(x)) is called uub-Gaussian with parameter o² if its Euclidean norm satisfies a sub-Gaussian tail bound:
Pr([ E[æ]]l e) S exx
VE>
(20)
20
2- Wasserstein Distance. Let μu and be probability measures on Rdd with finite second moments. The 2- Wasserstein distance between μ and v is defined as
(
inf f J3ax2"
1/2
W2(H,
||
2113 21
(21)
where T(u, denotes the set of all couplings of and v, i.e.,
T(,, v) = {7 E R²): r(A x Rd) μ(A), 7RR B) (BB) VA, B R measurable }
(22)
A.3 SUPPORTING LEMMAS
Tweedie's lemma establishes an important connection between the score of the marginal distribution and expectation of posterior when the likelihood function is gaussian. This allows the score function of the diffusion model to be used as a minimum-mmaan
aanniiuun--en---uuuu-----ne square-errurrr (MMSE) denoiser.
Lemma (Tweedie's 's lemma (Robbins) 1992.). Let po(aro) be the prior distribution and then De ~ N(xo, Σ) be observed with E known. Suppose Pt (xx)) be the marginal distribution of x.. Then, Tweedie's 's lemma computes the posterior expectation of Bo given a; as
E6 20 [.] = xz VV lo.:)
(23)
The lemmas related to 9-averaged from Combettes & Yamada (2055) are used to show fixed point convergence in [yy et al. (2019). In the following, we extend all these lemmas to a more general 5-weakly 0-averaged cases that will be used later show our ball convergence.
19

---

Published as a conference paper at ICLR 2026
Lemma 2. T: R - Rd be a function. Then, the following statements equivalent:
(a) T is 6-weakly 0-averaged for 2 and 0 E (0,
(b) [T(x) -(y)l 20)| =1l2 2(1 -)) T(y), y) S 880., for all
x, y RR.
(c) (1 1/0)1 (1/0)r' is ô-weakly nonexpansive.
(d) [T(x:) T(y)l 2 z || - - - -
-(1-))))))))(()()()))) -(T-T)(i)12++8) all IRd
Proof. Equivalence between (a) and (b): Provided T is 8-weakly 0-averaged, let's find the LHS RHS in (b)
(1 - 20) ull + [T(x) - T(y) 2(1 - 2) -1)))/y1100 y, (x)( T(y))
<(1 20)
a1l + [[R(x) OR(y) + (1-0) y)1l
2(1 () y, (R(()) R(y)) + 0)(x - y))
=(1
20) || 31ل (1))(()))) R(y) -1-)](| - -yl 2 2(1 0) a 211
+ 20 - (,,, R(y)) - 2(1 -0)0(x y. R(x) R(y))
y.
<(1
20) | 116 + 0| 001 (1-00))) yyl 2211 - 0) 211
+ 28(1 - - (()( - R(y)) - 2(1 0)0(= y. R(x) R(y))
=(1 - 20 + 0 + (1 0)2 - 2(1 0) 212 028
-0
For another direction, let suppose T satisfies (a). Let R $ (T-(1 -1 -1--0(10)) that have
T = 0R+ Now, we need to show that [|/(()) - R(y) 2 S || - y112+ for all x, E Rd
i.e. 8-weakly nonexpansive.
[R(x)) R(y))12
[T(x)) T(y) - - (1 - y)ll
02
02 (IT(x) - T(y)11 + (1 0)>1|> -v1 2(1 - ())(()) - T(y), x - y)
Ø2 (IT(#) - T(y)l1 + (1 20) v1l 2(1 (T(x) T(y), 8) 8 11
1
1
(088² + 1 =112)
02
=|[x y 1 82
where, the inequality is due to T satisfying (b).
Equivalence between (a) and (c): Note that is ó-weakly 0-averaged with Rt being d-weakly nonexpansive function. Now, we have
=
0R + (1 1)T
(1 11-1//0--0 1 (1/0)T (1 1/0)1 1/0 (0R (1-0)1
= (1 - 1/8) + R - (1-1/8)
R
Hence, T being 5-weakly 9-average is equivalent to (1 - 1/8)I + (1/8)T being 8-weakly nonexpan- sive.
20

---

Published as a conference paper at ICLR 2026
Equivalence between (a) and (d): From equivalence between (a) and (b), we have
[7(x)) - T(v) 11 + (1 -0) y112
2(1 - ())T()) T(y), - y) s 8202
- [T(x)) T(y) + (11-20]laa yl1
(1 (1-0) 0) (IT(x)) - T(w) + || - y 1 |(T (x)) - (T ()())1B)
S 620
- 0[T(()) - T 0|| - -yll - 8002 - - (TT -)(x)) (T - 2)))
IT(x) - T < x - -yl 0 [I(I - (x)x)- - - (T-T) +50
Lemma 3 (Concatenation of 6-weakly 0-averaged functions). Assume Ti Rdd - Red and T2 : Rd Rd are 51-weakly 01 -averaged and ó2-weakly Orzaveraged respectively. Then, Ti o T2 is 8-weakly O-averaged, with & -2, and 8² = //2 (6 + +830)).
Proof. Here, we follow the proof structure of Combettes & Yamada 2015). Since 01 - 02) S (1) we have 0 +02 < 1+0,02, and therefore, 0 -8, E (0, 1), and let 8² 620, + +820g
1-8102
0
Now, from Lemma 2 for i E {1, 2}, we have,
1 0,
[T](z) - (y) 11 S || U1ا
II(I (x) - (1 -I- 6000
(24)
Then, let us evaluate the composition function using this property.
Ti (2x Ti
(2()))1
< (2 - T2 1-0
- (T))(a) (T - (T)(((
0i
1- 02
<| x v1l
|[[ (x)(x)) - - (v)) 8502
02
1- 81
|(T - (T)(T((a)) - - (1) + 870
0,
From Bauschke et al. (2017) 2.15], we have, for a E R.
= 2ll + (1 (1- a1-a)| lau + (1 (1-a1 a) + a(1 a)llu aall- =12
- a) <
Now, let u = (I - (x))2=x)x= - (---)) -I- 1-02 and
1-01 b
6,
17 (x())) - Ti o (2))))
<| x v1l al|ua||
bl|0|
5701 + 8302
=| 21l
(a b)
a + b
|u 2 +
a + b |11|
+ 5701 + 1502
= x v1 (a ( b)
a b
|uu1 + (1- a
||0| + 00 + 550
a +
Using the above results, we get,
||T o T2(2)) - Ti o T2(y)
ab
<| v12 (a + b)
Iu + m0 + 870 + +8502
(a + b)2
ab
=||æ
21l
[I(T Ti (2) (T Ti o T2) + 8,0 + +8302
b)+
21

---

Published as a conference paper at ICLR 2026
Let f 21+00-20112 Then, that ab 1-0,02
1-0 1
0-
[[] T2(x)) Ti T2 (2) - x 2112
- (2) - (I Ti T2)
+
(25)
where, =²0= + 8200.. This implies that Ti T2 is ô-weakly 0-averaged with
0 01 02
20,02
1 o (600 8220)
8²
(26)
0102
Lemma (Proposition 5.4 of Giselsson (2015)). Assume is p-strongly convex, closed, and proper.
Then, -2Proxx p I) is ptp -averaged.
Lemma (Pardo 2018 ). The KI. divergence between two gaussian distributions q = =N(m, Σ1)
and Q2 = (12,, 2g) in TR.d space is given by
Σ2 Σ
d+ + (12 - 111) E5 (12 - mn)
Kl(g1 (142) log
(27)
where| || denotes the determinant, and denotes the trace of the matrix.
B INFLUENCE OF AC-STEP ON BRINGING CLOSE TO (Ma())
Given noisy image ((A) at each iteration k. the denoising aims recover the underlying clean image
+ (k),, where s(k) is the noise contained in ((h), The AC-step aims to bring 2(h) closer to the noisy distribution M, on which the sø (b1) was trained on. Lemma sssows that the AC-step tries match with the distribution induced by the forward diffusion process.
2; (k) ~ Po()) such that (k)) = :
(k)
Lemma 6. The KL divergence between the target distribution p(z (k) for correction steps and the distribution p( 2
(k) (z) induced by the approximate correction step Algorithm is given by
KL(p(zo(a) )0
(1(=(() (((0))
(ork))) 2 s(6)
(28)
Proof.
Target distribution: p (zow) (z(4)) = N
2/4
(k) (oci)) I
(29)
AC induced distribution: (k) ((()) = N
.(() (011)) (1)
(30)
The KI. divergence between these two distribution can be computed in closed form using Lemma
KLL(gll922) log - d tr() + (5(()) = =(²))T (a(t) -2
((2(k) = =(()))
d + d + (acii) -2 $1 =((2) 11:)
1
s/(t
2 (o(11)2
where, (kk) = x())_ z
Lemma 6 shows that KL-gap of our approximate AC update; as long as (k:) is sufficiently large, the two distributions remain close. Alkhouri al. Theoreem 1] showed result with similar
22

---

Published as a conference paper at ICLR 2026
flavor. Larger noise (A)) makes the posterior nearly indistinguishable, but it also washes out fine structural details originally present (low Signal-to-Noiss Ratio with larger o (8)). Existing works often use annealed scheduling o(k) + 0 (Zhu et al.
2023 Renaud et al. 2024a Wang et al. 2024) to 2 decays least as fast as (o(4) With just the
preserve image details, implicitly assuming
[[s(A]]
use of annealing g(k) schedule, it not sufficient ensure that = lands in desired manifold in each ADMM iteration. To bridge this gap, we propose to use DC-step in addition to the widely used annealing o (k) schedule that explicitly corrects this gap.
C PROOF OF THEOREM
The proof involves showing that the each iteration of ADMM-PnP is weakly non-expansive when the denoiser satisfies Assumption | of the algorithm.
This weakly nonexpansiveness of each step leads to ball convergence
Recall that the subproblems at kth iteration of ADMM-PnP is given by:
x (k+1) arg min 1 -((y|lA(x)) + 2- 2()) + (k)
Prox -Prox-ez -e(())
(31a)
(k+1) arg min 21(z)² + x: (k+1) - = + ck)
P Proxxn (x (++1) 1) + u())
Doc (x((+)) + (t))
(31b)
u (k+1)
(k)) (x.(+1) -((+1))
(31c)
Lemma 7 Ryyu et al.
2019). ). The steps of ADMM-PnP in (31) can be expressed as ulk+1) (v)))
with (R) (²) (R) and
T (2D ((t) - 12) Le -1) 1)
(32)
Lemma Do TRd - R&d satisfies Assumption
if and only if 112 - I) A-weakly
0-averaged with 0 = I+2c and A = 482 (1-0)
2c
62
Proof. We follow the similar proof structure as in Ryu et al. 2019). Let 0 = 2r
-+22 which implies
2.-05 Here, we can clearly see that 0 [0, 1). Let define G I+ze 12D 7) which implies D Dam 2(1-0)
1 G + IT Then,
[1D,,)) - 1)(x)= - (D - 1)))) - 2²1 212
=I1(D m(h) (x) DD - (x -y) 2 x y11
(1 Dock) 1 -111 (x) - (Doms) -
1
0
(4)
+ [ v11
x y
2
4(1 0)2
2( D Dow - 1 (x))
D Doam
(4), 2
(xc y))
4(1 0)2 |G((x) - G())112+ + 1 (1
6²
1- (1- (1 0)3
||0 212
2(1 - 0) (G(x) - G(y), x y)
4(1 4(1-0)2 (IG(z) G(v)1 -2(1 (G(x)) G(y), -0)) +(1 -20) -21())
23

---

Published as a conference paper at ICLR 2026
Now,
4(1 - 8)2 (IG(2) - G(y)l 2(1 ()((()) - G(y), x y) (1 20) x v11) - 82
1
++|((() G(y)l - 2(1 0)(((=) G(v), x - y) + 20) æ y1l < 48 0)²
+||((z) G(y)l12 2(1 0)(G(x)) G(v), y) +11 20) æ yll2 A²0
where, 2 462 (1-01² From Lemma 2
this equivalent to G being A-weakly --averaged.
C.I PROOF OF THE THEOREM
We follow the procedures in Ryu et (2019) and expand the results in the d-weakly expansive denoisers. We show that each iteration of PnP ADMM is also weakly nonexpansive when the denoiser satisfies Assumption
Proof. From Assumption
[(D - [)(x) -Dar0) - ()))) s y1 + 6²
(33)
From Lemma we have - (2Proxxe - 1) is - p+μ averaged.
Then, from Lemma g we have
(2D (1))) 7)
(34)
1 1 2c
8, wweklly 0-averaged with 0 = T-26
2r
and 09 = 462 (1-9)2
8²
By Lemma it implies
(2D -2) I]
(35)
1+ 2e
weekly 0. avveraad with 0. p+u++2pu
2pc and o2 8e 2c(1+26)*
18a
Now, using the definition of ww-wellyy 0. -eaaeerggssssss, we have
(2Dmm) (2Proxx - I) = - (1 2c) ((1 (1) 1 + R)
1 -(1 2c) p μ + 24c
+ 2μ1
R
p + 1 + 2/10
where, R is certain 6, weewly nonexpansive function. Plugging this result into ADMM PnP operator (Lemma we get
T
(2D -2) I)
I p+ 211e
(1+ 2c)
R.
p + + 241
p 4 2
P
(1+ + 2c) + 2 R
2(p + j + 2
2(p 4++ 2/16) b
where, clearly a > 0 and b > 0.
Now,
[[((=>) T(y) a²'la y + (()) R())l1
2(a(1 - V),((((z)) R(y))) (36)
From Young's inequality, for any > 0, we have
1
762
(a(æ b)) - R(y))) <
a²1
-11
+
|| R( R(y)l13
(37)
2"
2
24

---

Published as a conference paper at ICLR 2026
Plugging this, we get,
[T(()) - T(y)l12 S a² (1 1 7 | - 2112 + 62 +())) R(y)l
(38)
s d" 1 [++ (1+ + (1( || - b 7)6
(39)
where, the second inequality is due 8, weak nonexpansiveness of R. Note, that this holds for any 7 > O. When 7 = t. we have
a² 1+ (1+ + 6 02(1+))) = (a)
(40)
[T(x)) - 1
<(a + b)01(= - y1l2 + b²
1 a
) 8
p pe + μe + 2/162
2
(p + pe + jee + 2022
||0 - 81 2 +
(41)
+ μ + 2/e
c(p + μ + 2pe)
of
57
Hence, we have
[T(x) T T1)) f
216 +
oz
(42)
This shows that when eT
1, then, 3.- E Rdd and K 0 such that wk K the following holds:
(*) 0²
2
2
S
El
2k
0"
2
2
+
57
lim
(1)
2²
1
87
(43)
k0000
V1
E
Hence with this we have the sequence (b(() = z (k)
(k) JKEN converges within a ball of radius
8T
Since, -(2Prox - is ptp avveraged, this implies
1 (1- R)
Prox
(44)
2 2 1
for some nonexpansive function R. With this, have
( P )
lim
Prox (()(k))
Prox --rox 2o(0")
(0(0) 0"
k=000
87
lim 10000
(k)
x
<
(45)
pt pt 2
where x= = Prox y (="
With these results, we know there exists u* such that
lim 1t u² s (1
dt
(46)
p+ V1-
D PROOF OF THEOREM
Here, show that 3-step AC-DC denoiser satisfies Assumption for constants and 8. In the following, we first show that each step satisfies the weakly nonexpansive assumption. Therefore, the concatenation of these 3 steps meets Assumption 1
25

---

Published as a conference paper at ICLR 2026
Lemma 9. Assume the Variance Exploding (VE) scheduling (Karras et al. 2022 is used in the diffu- sion model. Given that the log density log Po (i.e. log Paata) a is smooth (Assumption the interme- diate noisy log- densities {log Pt M, smoothh for E [0, Т] iee. IV log (x) P: S Melk yll for all æ, y E X. For t such that (t) < 1/M, the smoothness constant Mr can be upper bounded as
M
M. S
S M
(47)
1 Mo²(()
Proof. From Tweedie's lemma, we have,
1 (² (0t E[ (x))
Vn, m, (og))())))
V2 0 Po
(²(t) (I V I3 2 (2,))
(48)
Now, let evaluate the Jacobian V Bl
V 2x
B[xolax] = Va.
zap(20o x; ddzo
p(az)
Do
V xi
po po(2r
Do
1
Va, (arp(ay(au)) p( (20) pf()))
Vas))(a)) po(
(o())))
JREE
(49)
Given 2, ~ (x0, (t) and p(æx)) = P(ao)(( we can compute their
gradient (similar to Peng et al. (2024) as:
V "2,
o²(t) p( x0)
V Va.P(D1) = / Po(200) V P(: oo daoo
JUNEX
po(20)
p( (20) ((e
-zo) dæo
BDE
d²(t) Pe (r - 2
(50)
Plugging these results in (49) and using integration by parts,
V(.p( p(xx))
2e, [x p(x.))
xo V (2
E(z0 [xe]
(51)
Substituting V Pe,p( (20) - 11/(0210)(2; x0))p(; (D),, we have
Vo,p(axe)
E pp [r,] = 0² (t))((z)) x0E
xo - mp p( (mo (oooooo))))))
12. (20 [20s]
m,
p(xx))
o2(()= E(z0 - x0)(2;]
p(xx,) E [01]
(52)
Substituting V p(.)( = Sa.Ex p(zu) V p (20 -P - (m.) 2u Po| we have
-1
V Eizn// æy] (t( E|zo( (x, xx) g²(t)
Ei x E|x, xoœ;]
o²(() Cov(xn, x; - x0)
1 Cov(xzalz.) (2)))
(53)
dt(
26

---

Published as a conference paper at ICLR 2026
Note that this result in (53) is similar to Dytso et al. (2021 Prropssition 1], and has also been derived Palomar & Verdu (2006).
in Hatsell & Nolte 1971);
From Assumption have
- -MI 3 V log 3 MI, Vao E X
(54)
where, M is constant.
Then, let's analyze the hessian of the log of posterior distribution p(z
V log p( (20 (xx) = V2 log + V log p (20)
(55)
1 2 logp(za)<;) = MI + o²(t)
MI+ + I=
(56)
(t)
mg
1
When M (1)
then the distribution p(ro strongly log-concave. In this case, the covariance of distribution p( ax)) can be bounded (Brascamp & Lieb 1976) as
200)
1
MI
Cov(2ng (0:))) 3
MI
(57)
o²((t
Combining result from equations
48
53
and 57p we get
1
V2 ax
lot)
(I
V
V.
B (x:))
(1)
1
1
Cov(
(2)
o2(1)
V2 10: S 0²(t)
(1 (7t))(
o²(1)
o²(t) Mo +1)
Mt²
o²(1) 1+ ((²(1))
M
(58)
T+ Mo²(1)
M
Hence, the smoothness constant of log g(xx)) is upper bounded as M, < 1 Mo?(!) i.e. Me s M.
Lemma 10. Let H =(k)) + Zac denote the function corresponding approximate correction in v,, the following holds for any x. E X
(k)
Algorithm 7
Then, with probability at least 1 - e-
( T)()) -HH(() n1w0 s (5(h)) 2 + +c((k) 22 y11
(59)
where, (k)² ( 22dvz + 22x),, and (c =0. 0.
Proof. For any x, y E X, we have the residuals ((l)) (x) = (H - [)()) = (4) nl and R!l (v) =
(H - (v) o (k) n2 where, nı1 n2 Lid. (OO
Then, we can bound the norm of difference of these two residuals as
((k)) (x: R2(k)
(1) ||3 | σ (x)
= =10(1))2111011))
(60)
27

---

Published as a conference paper at ICLR 2026
n1 12 2T) and xa is standard chi-square distribution with d degree of where, 7222 freedom.
From Laurent & Massart (2000) [Lemma 1], the following holds with probability at least 1 e
Xa - d + 2 duv + 2/h
(61)
Plugging this in proves the lemma.
Lemma 11. Let H( zac a denote the function corresponding to fine correction as defined Then, with probability at least c e-v the following holds for any æ. E X if 0 a(n) Me.
(k)
in Algorithm /
111 - ()) -HH - -1)(8) - (o)e)) + (Ede (2 -y11
(62)
where, (
32dor ((0) log 2, and )= (1- ( ux
12MM 21k2
1- (1)) Mr
Proof. Recall that the target distribution for this step is given by
logp(zzcu (x(k)) log p ((=(01) (z,(0) (2,()))))))))))))))))))))))))))
log p(z)(1))
(63)
(l) where, p(zac (0())) (2 I).
2 and 3 the target distribution p(z,00) (z(e)) also inherits smoothness and coercivity properties. These conditions imply the ergodicity of corresponding Langevin diffusion
Under Assumptions
(Mattingly al. 2002 Chen al. 2020 In partieular, Bokker-Plances equation (Uhlenbeck &
Ornstein 1930) characterizes p(=a()) Zacc as its unique stationary distribution. Consequently, the
de obtained through Langevin dynamics converge to this distribution as the step size (7(k)) - and the number of iterations J oc. The gradient and hessian of the log of this desired distribution are given by
iterates
(k)
log (2 x
nc
(N)
de
(k)
V (1( zdc
log Pe (z (1))
(64)
((R)
2
log pl
(k))
2 de
log Pt (z
(65)
de
dr
(k)
Here, / refers to the noise level such that (1)) (k), By the M--smoothness of log pt distribution (Lemma 9. we have
Mr +
log (2 1=
(R) Mr
)2(m)) (
(66)
(k)
When Mj <
the hessian is negative semi-definite that implies the distribution being log
.(N)
concave. This also implies that when M, << the likelihood term dominates in the posterior
63
Using
Wainwright,
3.16]. the following holds with probability least 1 V/ (k)
de B[
(k)
S V
log
2
(67)
X
2/k
where >₂ -M,
(00)
Now, using Tweedie's lemma, we have
E 2 [= 2 + ) V
(k) (k)
(k)
(2) log p( p(z(6) (k))
E 1
(k) [k) z
= (2)))
(1) log p (K)
(68)
28

---

Published as a conference paper at ICLR 2026
Combining the above two results, with probability at least 1 - 2/k, the difference of residual for and y can be bounded as
n (a:
R/
(
2
2
4
log
2
+ 11002 (Vo log
Vy
log (1) (v)) (w)) ||
2
Vk
suc
32d
log
2
2
0211)
(V Vollop_.mm (x)) - Vy log (0 (v)) ||
X
2k
32d
2
log
+
20
4 B(N)
M
(2200) ||
y 1
(69)
d
Uh
=0o
where, M Ma (o) is the smoothness constant of log logp_00
a0 that can be derived using a similar proof
procedure as Lemma D and d = M +
Following the similar procedure as in proof of Lemma
9 we get,
Mt
M Zuc
(70)
Mc
Plugging this leads to the lemma.
Lemma 12. Let (k) (k) in Algorithm 7
Zae (k) denote the projection function using Tweedie's 's lemma defined Then, we have the following
11T [)(x)) (F. ()vi) )11x
--12 -5
(k)
(71)
for any x, E X. where (<tw (1))))2 (o(8))* and 0(8)2 O.
Proof. From Tweedie's 's lemma, we have
(k)
B[ 2t = zdc [k))
de + (a ( (8))
(k)
where, / E [0, 7] such that (1) = o().
Then, the residuals are given by
B!() (x) (2) 2 log (z) n (y) (o (k) 2 log pr (3)
(72)
(73)
Now, the norm of the difference of the residuals can written as
R (x) R(W (v) || = (6 log pr - V 1og
(74)
(o( pr -
(75)
where, Mj is the smoothness constant of log (x)
L
D.1 MAIN PROOF:
Proof. Part (a) Using Lemma 10 11
and 12 with probability at least 2e 2e-", we have
RBa (x) R (y)
2
<3 [x)) - R(()) (y)
3 R (x)
(k)
(v)
3 (k) (x) (R) (2)
(1))2 + (c(
(k) 2 + (1) 1))
211 + (()(k)2
+5(h)² 22)
+ ((4)2
+ do
ac
< x -v2 02
(76)
29

---

Published as a conference paper at ICLR 2026
where,
=3 (
((V2Moois )²
2u, Me + +
32d0 k(+)
82 = 3 2(((0))) (d 2 du. + +2k) + (1 Mio a(a))
log
Using Lemma 9 leads to the final theorem.
Part (b). Let us Pa In
2x
2 In k. With this, the above weakly nonexpansiveness holds for
all l E N with probability at least
In
2n
-
21n
2c
1=
6n
(77)
k=
Using Riemann zeta function (Titchmarsh & Heath-Brown 1986) at value 2, we have
1
π²
((2)
(78)
22
6
k=1
Plugging this in we get the probability to be at least 17. Now, combining the results with Theorem leads to the final proof of this part.
E PROOF OF THEOREM 3
Here, show that our 3-step AC-DC denoiser bounded with high probability. We first show that cach of 3 steps are bounded, and then combined them to establish the boundedness of our AC-DC denoiser as a whole. And following the boundedness, we show that AC-DC ADMM-PnP converges to fixed point with proper scheduling of (k) and oglk).
Lemma 13 (Uniform score bound). Suppose Assumption 2 holls. Let
D dia sup | Ull2 and S = inf =.yEX
log Pdaza (x))l22 (x))l oc.
Then, with L, MD + S, we have
sup #EX
V log Pditta L.
Proof. From Assumption
2
we have
V log (x) v log Puata S M | - 81l2. Va, X'
(79)
Fix any X. By the triangle inequality, for all æ E X
V log Pataaa (x)))
V log Pdasa (x)
V log (2o))l2 + log Pdata (xo)ll2
< M || æoll2
V log ( (x))l2
(80)
Taking the supremum over a X and then the infimum over Bo X yields
sup V log Pauta Pa)ata()))1 S sup M| zoll2 + inf |/ log Pdata (2o)) ll
(81)
xEX
=eX
#xE
< MD S
(82)
Because ||4]|oo < |/ulll2 for any vector, sup
V log Pdata (2) S MD + S
(83)
#Ex
which proves the above lemma.
30

---

Published as a conference paper at ICLR 2026
Lemma 14. Assuming V log Pdata (x))ll S. Vx E X, the score of intermediate noisy distributions (pr)eeto.T) are bounded as.
[Vlgga S VaL
(84)
for all X.
Proof. We have I2 zo + (t) with ~ (0, I). Let us denote n2 = at)m.. Then, the marginal distribution is given by the convolution of two distributions.
(x(x)) = [
Pni (01)po(2 x² x2)de
(85)
2IEX
Then, the score is given by
(x)
Vlogp((=)
(x
1
Pni (21) - x, 11)
Po(x))
1
Pni Pri (2:1))P( a1 V Po x11 day po x1)
Po(æ))
JIE
1
Pei Pmi (21)pol 21 V logpolæ a dd)1
(86)
pr(x))
Now the norm can be bounded as
(o -
Pni (x1)po(< -- V logpo(x - x1)11,dx1
Pt (x) DIEX
(e( [
< sup miEx
V (og
Jaiex Pni (o:)01o x )dx1
= VdL
(87)
The final equality is due the fact || I Vdllollo..
Lemma 15. Let H(l)) (2()) + zae denote the function corresponding to approximate correction to noise level (k) defined in Algorithm for any x. y E X
Then, with probability least 1 -- the following holds
( - ( (1
(88)
Proof. For any x X, we have the residual (k) (x) (Ha (k) 1)(x)) (k)), 1). Then,
R (2x) (a:) ||- (0 (a (x)
(89)
where Xa is standard chi-square distribution with d degrees of freedom. From Laurent & Massart [Lemm) 1], the following holds with probability at least -M
x $ d 22 + 2v
(90)
H I=(3) < (1) 2/ + 2v) with probability at least 1 - - due
This implies to d 2 1.
Lemma 16. Let (1)h) zuc 2 denote the function corresponding fine correction defined in Then, with probability at least - -" the following
(k)
(k) (k)
Algorithm [ Assume |/ log Paata (x))l S holds for any x. y X:
(H())) de
()x))
1/3 S
8
2 log + 0 I,2
(91)
A
where, X, -M,
200
31

---

Published as a conference paper at ICLR 2026
Proof. From 667)
with probability least 1 2v, the norm can be bounded
R(
2 -
2 + ) V log Pogp 0o (a:)
(x)
log
(92)
X
v
Then, using Lemma 14 we have
2
2 log
d de
(x)
I
+ 0 g00 2
(93)
d v
Lemma 17. Let H
2dg + (1) denote the projection function using Tweedie's 's lemma defined
(k)
in Algorithm 7 Assume V log Pdata(a:))l S L. Then, we have the following
d ( 1)(20) (0(())=
(94)
for any X.
Proof. From Tweedie's lemma, we have
(A)
E[ 2t 2dc
Ftw
(8) de
+ (a (0(0)) Vog (K)
(95)
where, E [0, 7] such that ((a) (t).
Now, the norm of residual can be written as
R
kk)
| - (0 [lg
(96)
where, L bound of gradient from Lemma 14
E.
MAIN PROOF
Combining Lemmas 15
16 and 17 leads to the proof of part (a) of Theorem
With probability at least 1 2e "k, the denoiser satisfies the bounded residual condition.
l/dl((Dow: - 1)(x)11 - ck
(97)
Let's define the relative residue as:
1 Bx Va (100
x (k-1)
+ 2((() 2k1-1) ||, + -
-
(98)
For any 7) 1) and constant 1, the penalty parameter Pk: is adjusted each iteration according to following rule (Chan et al.) 2016):
Tpk
if Bb+1 2 n (Case 1)
Pk+1
(99)
Pk
else (Case
The PnP-ADMM with adaptive penalty involves two cases as shown above. At iteration k; if Case holds, then by Lemma 18 we have
2R
But1 < 6ck + 2cx
(100)
Pk
On the other hand Case holds, then,
Bk+1 S n1
(101)
32

---

Published as a conference paper at ICLR 2026
Define 0ę 6c; + 2ck-1
28 Pk
Combining two cases, get
?7 0, Case holds at iteration k:
if Case 2 holds at iteration k
3e+1 S 6,3g an 8
(102)
Note that for Case Pk+1 7Ph and with 7 > we get limg- c.
In addition, with
Pk = In 2r 6n
+ 21n k, and the scheduling of o ((*), (A)) that satisfies
lim (o (1())) 1V + 4v%) = 0, lim
244
log = 0, lim o (8) = 0, lim x
k-+00
k-+00 1 2
Uk
k0000
1-+000
results in lim a₁ 0.
As + oc, 3 different scenarios could occur. Let's analyze each of the scenarios one by one.
Scene I: Case I occurs infinitely many times and Case 2 occurs finitely many times.
When Scene occurs, then, there exists constant K₃ 0 such that B&+1 ag for all k 2 Kj. Since limg ag = 0, this leads to lim,->se 3x = 0.
Scene 2: Case 2 occurs infinitely many times and Case occurs finitely many times.
Similarly, there exists a constant K2 > such that 3x+1 S nB for all k > K2. Then, we have
Bx S 1 3K, And with 77 [0, 1), we have lim 8x 0.
(103)
Scene 3: Both Case and Case occurs infinitely many times. With the Scene I and Scene 2 converging, the sequence lim3 Bx 0 under this Scene well.
This proves the part (b) of Theorem 3
Lemma 18. For any iteration k that falls into Case 1, the following holds
2R
Bat1 S 60g + 20x-1 +
(104)
Pk.
Proof. Consider the subproblem 7ab with adaptive penalty parameter Ps (as defined in (99).
x
(b+1)
arg
min ((vll + 2
(R) + (k)
(105)
Pk
With the first order optimality condition, the solution x (k+1) satisfies
V (=<(y\((xx))
(==((+1)> + (x(++) zz()) (K) + (A()) =0
(106)
1 Vd
x (k+1) (k) + u (b)
(107)
PLVd
Using the assumption of existence of R < oo such that get
Vo X. we
(k++)) (b)) + (k)
|| S R
(108)
Since the denoiser Daas is bounded with probability least 2e- have
11, ()(x))112 -
(109)
Now,
x (k+1) (k+1) + r (b)
=
(k+1) + (k) - Dovs (2 (H+(1))) + (0))
(110)
Vil
v/2 (Docs) ()(x(++1) + 20 |
(111)
S ch
(112)
33

---

Published as a conference paper at ICLR 2026
Now, using triangle inequality, bound || ||=(1++1) - -2(0)/1 (k+1)
x(++2)
x((-11) ((() + (k+)1) (k) 2(0)
(113)
Vd
R
Ck
(114)
Pk
Similarly, it can be shown that
d
u
(++1)
2
=
vd
(k) + (b+1) - 2 2((t1)
(115)
1
(a) + x((+1) - Doris (2) u 20/
(116)
Vd
1
(Dams) - (2((+111) + 3
(117)
Vd
(118)
This implies 1
(1++1) - (x) 112 S 2ck- Finally, use (+++1) (k++1) - (R) + (k++) to
Va
obtain
(k+1) =0(1//
(119)
1
(H+1) (A)
+ x1++11 (A) + (k-1)) - =x(
(120)
Vd
2
$ Vd
1
u (R++1) u (k) =110111
1
(A) -l 11/ + vd
(R+1)
(b)
(121)
vdl
R
<2ck
+ 2ch-11 1
+
+
Ck
(122)
Pk
R
=3c + 20k-1 +
(123)
Pk:
Combining all the bounds results using triangle inequality results
2R
B&+1 S 6ck + -2kk- +
(124)
Pk
where, C = (22 4/Dr + 4vk) + (60"(ts/1-Mo)) log?/
2/o, + 20 sln) L2 + 2(0(00)44
Remark 1. While the proposed method and its theoretical results are based on Variance Exploding (VE) scheduling, they can be easily extended to Variance Preserving (VP) scheduling case (Karras
el al. 2022).
E.2 THEORETICAL RESULTS WITH FINITE DC STEPS J
denote the function corresponding to fine correction as defined Lemma 19. Let H(()) (1) + 2 dc
(k)
in Algorithm I wwith finite J and (9(k) S 20 Also, let (k) p((z00)) (k), be the stationary target distribution and *(*²) be initial distribution used for the DC at iteration k. Then, with probability at least 1 - e - the following holds for any x, y E X 1/ < Me
(H -I)(z)
(H
(R)
1(3) 112 < (5(k))) + (c
(k) (0))) - -11
(125)
w < < 1, C 0, (5(4))2
64do²
log + C(1 - k3) 2/ W x (00) +0 ((n))))2).
M-- (())
and (b)22 22/2M00211)
34

---

(k) Published as a conference paper at ICLR 2026
Proof. With finite J, the langevin dynamics doesn't necessarily converge to the stationary distribution (k) p( Zac ). Let # be the initial distribution used to initialize the finite step langevin dynamics and ((k) be the distribution the iterate after running finite J steps of langevin dynamics. Using (Dalalyan & Karagulyan [ 1], the following holds for (x S 22 8(4)
W2 xx)) & (1 n:) (M) , (b)) + O (nrat)
(126)
where, and W2 is the 2- Wasseerstein distance.
Using Kantorovich and Rubinstein dual representation (Villani et al. 2008 )
[[ [z] - B, [z]122 - ///
(k) de dif (x
/ zdm (1) (2)
(127)
12
S Wi (++(), ,x()))
(128)
S W (x)))
(129)
The last incquality is due to the Holder's inequality Villani et al. (2008). Using triangle inequality leads to following:
100 [z]]
1=11|=
S
Egt0 [z]]
+ -| [z] E aa [=]1|2
(130)
-
de
(R)
( [x]]
+ +(1 xz) m)) 7 (b))
+ O (7763)
(131)
Using this result and following the same procedure as in Lemma 11 we get the theorem.
Theorem 4. Suppose that the assumptions in Theorem 7 Assumption 2 and Assumption hold
Further, assume that the DC steps finite steps J and rh - 20 20 Also, let (k) p(z Zan be
the stationary larget distribution and =[K) be initial distribution used for the DC at iteration k. Let
Does, ( ((1) denote the AC-DC denoiser. Then, we have:
With probability at least 1 - 2e_ the following holds for iteration k of ADMM-PuP:
,!(D,,.) [)(x)= (D,()) - () - द y112 + +
(132)
for any y, E X. k E Nt. constant 0 < K < 1 and C > O. when 02.0, + (a(k) < 1/M with
2f = (122220014)/
)) (+))
(133)
82 3(2(( (2) 2 du + 2vx) + B4do Mo? ((1)) log 3/ve+
C(1 - x) 2J W (1) 7 (b)) + +0 ((n11))2).
(134)
In other words, with 12 = In 27/61 + 2 In k. the denoiser Dook) satisfies part for all l e N+ with probability at least 1-11 1).
Proof. By substituting Lemma with Lemma 19 leads to the theorem.
Theorem 5. Suppose that Assumptions hold. Let diam(X)) sup yEx ||0 U1l2 <
00; S : infaex V log Pasts(())l < Dc and define I= MD + S. Let Dacms ((k) z denote the AC-DC denoiser. Further, assume that the DC steps finite steps J and (R) S 20² 20? Also,
s(n)
let (k) ((2 2(0) be the stationary target distribution and =1)h)) be initial distribution used for the DC at iteration k. Then, the following hold:
(Boundedness) With probability least -2- ue, the denoiser Dacss bounded each iteration 1 - S of whenever 0².0) + (k)2 < 1/M, where Cs (4)) 4 + +4x)+ 320" +200 3200220
l i.e.
log 2/v2 CC1-x)) (=(()) (k))) (LL))
4
( 0 < < 1, C 1,
0 and Us > O.
Le In 2x² + 22 k with n E (0, 1]. Consequently the denoiser Docer is bounded for all k N. with corresponding ex and probability at least 17.
6n
35

---

Published as a conference paper at ICLR 2026
Proof. The proof follows similar as in Theorem B incorporating the effect of finite J in Lemma 16 as done in Lemma
F THEORETICAL RESULTS FOR ODE BASED DENOISER
Refer the Zhang al. (2024) for details ODE based denoiser.
F.I THEORETICAL RESULTS EQUIVALENT TO THEOREM 2
Lemma 20. Let H(le
dec
(M)
2
ode denote the projection function using ode based denoiser
(k)
Karras el al. 2022 in Algorithm
7
Then, we have the following
|1A l)(x)) - -H - - S (code)
2112 + +
(135)
for any I, y E X' with (code) == = ( ( dt) and ((())
Proof. Then, the difference of residual ode projection i.e. R H((h) ode
can be bounded as
2
0
(a)) - R (1)
(V) V (log):()))d
(K))
-o(()o(()(( (ogg(()))))))))))) V log p. (1)) (d)
1=1 .00)
2
(o(t)o'(t)) (Vlog))()) V Po
22
(a(t)o'(t)) M - - y11at
22
(a(t)o((t)) Midt || -311
(136)
Theorem 6. Suppose that the assumptions in Theorem Assumption and Assumption hold.
Further, assume that the step size satisfies 1/(h) - O and the number of iterations J oc. Let
Daus ((k) 2(0) denote the AC-DC denoiser. Then, we have:
(a) With probability at least "k the following holds for iteration 1: of ADMM-PMP:
[1(),m) - 1)(x)) (Dam.) - -1)())11 s &
yll +
(137)
for any x, y E X and K E N+ when 20 + (0(1))2 < 1/M with
t 3(V2M M)2 + 1"
(It)) Medtt)
(138)
& = 3(2 (1) 2/ dus - + 22x) + 32d0 (00/ No² (0r)) log 2/vn).
2/
(139)
In other words, v = In 20/6n + 2nk, the denoiser Daw: satisfies part (a) for all k E N+ with probability at least 1 n
(b) Assume that (k) is scheduled such that limg (a (k) 0 vk = In 2x /6n + 2nk, 1, and f/u(1+e-22 < 1/8 all hold, where = limg- sup with defined in (38). Consequently, limg-soc sup is finite and ADMM-PnP with the AC-DC denoiser with ode based denoiser converges to an r-ball (see in Theorem 1 with probability at least 17.
Proof. The proof follow similar to the proof of Theorem in Appendix D.I with the residual bound of Tweedie's lemma replaced by Lemma 20
36

---

Published as a conference paper at ICLR 2026
Table Hyperparameter settings for each task
Task
p
W
of Adam in 7ap
Superresolution (4x)
100
100
3 x 10
Gaussian Deblur
100
100
5 X 10
HDR
500
100
3 X 10
Inpainting (Random)
500
100
1 1 10
Inpainting (Box)
500
100
1 X 10
Motion Deblur
100
100
1 X 10
Nonlinear Deblur
300
400
3 X 10
Phase Retrieval
100
400
1 10
F.2 THEORETICAL RESULTS EQUIVALENT TO THEOREM
Lemma 21. Let H
ode
2 Zode denote the projection functiin using ode based denoiser
(k)
Karras et al. 2022 in Algorithm
7 Assume |/ log Pdata( S L, Va E X. Then, we have the
jollowing
Ma)) $ Sr (o(t)o'(t)att
(H(()
(140)
d ode
1=
for any x X.
Proof. Then, the residual of ode projection i.e. R(()
H
I can be bounded as ode
12 1
|| n
(ar) (az)
0 (1)o (1)V log Pe (ar)
|= ((k)
Sd. T
(t)o
1=t (00)
Sd 1,2
(o)) (t)dt
1=t (00
2
<d. I
(t)t)
(141)
( (M)
theorem analogous to Theorem can also be obtained ODE-based denoiser. The only difference lies in the expression of the constant cc, which in this case becomes
1
c = (2 + 100 Mo log 2/22 + 20 L2 +222
(D(1)o'(1)
1=
G USAGE OF LARGE LANGUAGE MODELS (LLM)
An LLM was used solely to assist with polishing the writing. LLM played no part in the experiments, results and conclusion.
37

---

Published as a conference paper at ICLR 2026
H EXPERIMENTAL DETAILS
H.1 DETAILS ON TASK SPECIFIC
LOSS &
We use mean square enoor (MSE) as the data-fidelity loss for every task i.e.
1
[(y][=) = log log p(y)z)
20 11 - A(>)
(142)
H.2 DETAILS ON PRETRAINED DIFFUSION MODELS
The pretrained models provided in Chung et al. (2023) are used in our experiment. Refer to Chung al. 2023) for more details these pretrained models.
H.3 BASELINE DETAILS
Unless mentioned otherwise, we conduct the experiments in default settings of their original implementation except for maintaining consistency within the measurement operators.
DDRM (Kawar et al.
2021):
We use 20 steps DDIM with v 0.85 and T/b as
specified Kawar et (2022).
DPS (Chung et al.
2023
The original implementation ran their default settings.
DitPIR (Zhu et al.
2023
The default settings are adopted in the experiments.
RED-diff (Mardani et al.
2024):
We use 0.25 and Ir 0.5 as specified in the paper.
DAPS (Zhang et al.
2024
We use the best performing DAPS-4K version proposed in
the paper.
DPIR (Zhang et al.
2022): : We employ "drunet_color" as PnP denoiser, while keeping all
the other settings at their default values.
DCDP (Li et al.
2025): All the settings are set to their default values.
PMC (Sun et al., 2024): : PMC was proposed using different score models for two different
tasks with relatively high measurement SNR. For fair comparison, used implementation with the same score model checkpoints as our methods, and further tuned this method accordingly.
H.4 EVALUATION METRICS
For all the methods, we use the implementation PSNR, SSIM, and LPIPS provided in piq python package. The default settings for these metrics used except the average pooling enabled for LPIPS.
H.5 COMPUTATION RESOURCE DETAILS
All the experiments were run on a instance equipped with one Nvidia H100 GPU, 20 cores of 2.0 Ghz Intel Xeon Platinum 8480CL CPU, and 64 GB of RAM.
I
ADDITIONAL EXPERIMENTAL RESULTS
I.1 ILLUSTRATION OF PROPOSED DENOISER
Figure 6 illustrates the effect of the proposed correction-denoising procedure. The noisy input image typically lies far away from the Gaussian noise manifold, leading to poor denoising performance if directly used. To address this mismatch, method first performs correction to effectively gaussianize the noise which is then denoised using Tweedie's 's lemma or ode-style score integration, producing high-quality clean reconstruction.
38

---

Published as a conference paper at ICLR 2026
16
100
200
100
300
200
250
Noisy Image 200
Corrected Image
Denoised Image
D (200(2
Figure 6: Illustration of correction and denoising step in proposed method.
Histogram of Plog Dasa (xs) Viog Pars lxolll
[x: kib
80
Frequency
60
20
120
140
200
Ratio
Figure 7: Histogram of score difference norm ratio: |/ log Prtara (x2) - V log Printa (21))12/112z
22l12. illustrating empirical smoothness (Assumption 2).
1.2 EMPIRICAL VALIDATION OF ASSUMPTION AND ASSUMPTION 3
To assess the practicality of the smoothness and coercivity assumptions used in Theorem 2, conduct two diagnostic experiments using a pretrained score model on the validation split of the FFHQ dataset. These experiments are designed to evaluate (i) the empirical Lipschitz behavior of the score function V log Pdata (z) (Assumption 2), and (ii) the coercivity of the energy landscape log Pdata (x) (Assumption
Empirical smoothness of the score. We randomly 1000 samples of æ Ø2 and compute score differences || V log Paata (x11 V log (x2)l and image differences || - -x2ll2. Figure plots the histogram of their ratio. The distribution concentrates around finite value (mostly between 50 and 160), indicating that the score behaves approximately M-Lipschitz with moderate empirical constant. This supports the smoothness requirement in Assumption 2
Empirical coercivity. To evaluate coercivity, we scale images by factors E 1.5, 2, 3] and measure the quantity (ar, - V log Paata (x:) as function of the squared image norm ||>|| As shown in Figure 8 the inner product grows approximately linearly with ||>]| indicating that the learned score consistently pulls large-norm images back toward the data manifold. This behavior is consistent with the coercivity structure assumed in Assumption
Together, these empirical diagnostics demonstrate that the theoretical assumptions employed in analysis hold approximately in practice and therefore justify the use of DC correction in our AC-DC algorithm.
39

---

Published as a conference paper at ICLR 2026
Empirical Coercivity Test
1e6
Voog
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1e6
Figure Empirical coercivity test: relationship between (ar, V-VVlog log Pdata (a:) and || The strong
positive correlation indicates coercive energy behavior (Assumption
Table 4: Reconstruction metrics (100 images) on FFHQ ImageNet for additional tasks. Bold: best, blue: 2nd best.
FFHQ
ImageNer
Method
PSNKT
SSIMT
PSNKT
SSIMI LPIPS.
Ours-twasdic
27.425
0J853
0.164
26.515
0.817
0.182
DAPS
26.94
0.852
0.154
26.845
0.816
0.172
RED didf
26.815
0.836
0.241
20.794
0.771
0.232
HDR
PMC
21,582
0.700
0.291
22.745
0.707
0.290
Ours tweedie
29.326
0.823
0.185
27.837
0.725
0.212
DAPS
28 28.598
0.781
0.172
27.245
0.739
0.201
DPS
23.746
0.668
0.276
22.724
0.543
0.394
Nonlisear Debiar
REDDdiif
26.9
0.77
0.234
25.488
0.77
0.207
PMC
21.102
0.0023
0.354
22.547
0,533
0,430
1.3 ILLUSTRATION OF USAGE OF ADDITIONAL REGULARIZATION
To further demonstrate the flexibility of integrating diffusion-baset PnP denoisers within the ADMM framework, we present an example where we employ an additional perceptual regularization term which will be handled in the maximum -ikullm--ikllhoodd (ML) step. In particular, the x-update step of ADMM with an LPIPS perceptual regularization (Zhang et al. 2018) becomes:
x (k+1) arg min [((llA(x)) + -11== (1)) + "* + Appips LPIPSyco Tret); (143) #
where Fref is the reference image and Appips controls the perceptual strength.
This example highlights the flexibility of the proposed method: unlike traditional diffusion- -bsed PnP presence dual variables, our design enables seamless incorporation
approaches that struggle of additional regularization terms. In Fig. 9 we illustrate box inpainting reconstruction task with the perceptual LPIPS- VGG regularization which enhances semantic content consistency while allowing visual style transfer from the reference images.
1.4 RESULTS ON ADDITIONAL TASKS
The results additional two tasks: HDR and nonlinear deblurring presented in the Table 4
40

---

Published as a conference paper at ICLR 2026
Reference Image 1
Reference Image 2
Original Image
Measurement
Reconstruction with Reference Image 1
Reconstruction with Reference Image
Figure 9: Demonstration of incorporating additional perceptual regularization.
Table 5: Comparison of our method with and without correction FFHQ. Best results highlighted in bold.
Ours-tweette without currection
Ours- with currection
Tisks
PSNK?
SSIM
LPIPS
PSNRT
SSIM1
LPIPS
Superresolution
Canssian
Inpoisting (Box)
Nonlinear Deblur
21.731
Retrioual
11.978
I.5 ABLATION STUDY
We perform the ablation study on the significance of our proposed correction steps. The results are presented in the Table 5
1.6 INFLUENCE OF DECAY SCHEDULE AND NFE EFFICIENCY
In our ADMM-PnP scheme, the size of the decay window for (kk) determines the total number of iterations and thus the speed of convergence. A shorter window (small W) drives (k) down more quickly, often reaching convergence in fewer steps but at the risk of settling in a suboptimal local minimum.
To study this trade-off, sweep
W E (5, 10, 50, 100, 200, 300, 400, 500)
for each task. Since each iteration of Ours-tweedie and for the Tweedie's lemma based denoiser). these W value translate to
11 score evaluations (10 for the DC update
Number of Function Evaluations (NFE) (55, 110, 550, 1100, 2200, 3300, 4400, 5500)
41

---

Published as a conference paper at ICLR 2026
DOVOR
0..0
27.
Dus Seaeci
141
127
Figure 10: Performance with respect NFE for Superresolution task (FFHQ)
080
CUEESOO
083
DAPIS
0.1
DoM
00
032
Figure 11: Performance with respect to NFE for Gaussian deblurring task (FFHQ)
3.1
20.
Caseer
DIPS
hes
11A
Figure 12: Performance with respect NFE for Inpainting with random missings (FFHQ)
By contrast, each Ours-ode iteration costs 20 NFEs, giving
Number of Function Evaluations (NFE) {100, 200, 1000, 2000, 4000, 6000, 8000, 10000}
Figures 10 15 plot mtan-std. (standard deviation) performance of our methods and all baselines against NFE over 100 images of FFHQ dataset. For most tasks, quality saturates after just 10 iterations (110 NFE for Ours-tweedie, 200 NFE for Ours-ode), showing a rapid decay schedule suffices achieve near-peak results. However, on the hardest inverse problems (phase retrieval and nonlinear blur), gradually decaying noise (larger W) and more NFES yield significantly better reconstructions-Fn far outpacing every baseline. Thus, while aggressive schedules excel on simple tasks, challenging problems benefit from extended iteration and gentler annealing; given enough NFEs, our approach establishes state-of-the-arr performance across most of the tasks.
1.7 MORE QUALITATIVE RESULTS
42

---

Published as a conference paper at ICLR 2026
0.56
OPEEW
Da
0.5
LKV
Figure 13: Performance with respect to NFE for Inpainting with box missing (FFHQ)
0455
QUEFOOE
PAP
0.5-
0,00
034
Dare
0029
9/4
Figure 14: Performance with respect to NEE for Motion blur (FFHQ)
Hil-ge
SDDF
Figure 15: Performance with respect NFE for Phase retrieval (FFHQ)
43

---

Published as a conference paper at ICLR 2026
Original
Measurement
Ours-tweedie
Ours-ode
DPS
DiffPIR
DAPS
DDRM
Figure 16: Recovery under 4x superresolution task FFHQ
44

---

Published as a conference paper at ICLR 2026
Original
Measurement
Ours-tweedie
Ours-ode
DPS
DiffPIR
DAPS
DDRM
Figure 17: Recovery under inpainting with random missings on FFHQ
45

---

Published as a conference paper at ICLR 2026
Original
Measurement
Ours-tweedie
Ours-ode
DPS
DAPS
Figure 18: Recovery under motion blur task on FFHQ
46


Reviews:
[{"review_id": "8Iy6Fst2wQ", "reviewer": "Reviewer_uYpW", "summary": "The paper proposes a new plug-and-play (PnP) ADMM method using score-based priors. Since score networks are trained on noisy manifolds, while ADMM iterates (due to dual variables) may not lie near those manifolds, score-based denoisers perform poorly, and the convergence theory is unclear. To address this, the paper proposes the AC–DC denoiser inside ADMM, consisting of three stages:\nAC: add Gaussian noise to push iterates toward score training manifolds;\nDC: short Langevin update guided by both the score and a quadratic prior; and\nFinal denoising via Tweedie score formula. \nUnder the strong convexity of the data fidelity term, the paper shows that the ADMM iteration is weakly nonexpansive and converges to a fixed point (up to a $\\delta$ ball). Experiments on several inverse problems (inpainting, deblurring, SR, phase retrieval) show improvements over PnP baselines (DiffPIR, DDRM, DPS, RED-diff, DPIR, etc.).", "questions": "- The DC step convergence assumption seems very strong. In practice, you run a few steps; how sensitive is the method to DC iteration count?\n- How is $\\sigma^{(k)}$ chosen in the AC step? Is there a principled schedule, or is the choice purely empirical?\n- Can the method diverge with fixed $\\rho$ for non-convex cases? Any empirical failures?\n- Can you provide some empirical evidence that dual variables distort score manifold geometry?\n- Does AC–DC satisfy a consensus Equilibrium interpretation?", "limitations": "", "rating": 4, "confidence": 2, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "- The paper addresses a real gap: score-based priors in ADMM are harder than proximal-gradient PnP, mainly due to dual variable noise geometry.\n- The method conceptually simple: AC noise + short Langevin DC + score denoise.\n- The paper extends PnP-ADMM convergence analysis to score models and Langevin steps. \n- Broad experiments across common inverse problems show the practical utility of the method.", "weaknesses": "- Noise addition before score application already appears in recent PnP diffusion works (which the paper also acknowledges). In this respect, the novelty seems incremental. \n- The DC step, a short Langevin refinement combining the score and a quadratic potential, structurally resembles diffusion posterior correction schemes, where noise injection is followed by brief score-based Langevin updates to pull iterates back toward the learned distribution. The paper’s approach integrates this idea into ADMM, but conceptually overlaps with prior noise-add-and-refine mechanisms.\n- The AC and DC schedules appear heuristic. The issue of stability is unclear across tasks without tuning.\n- Theorem 2 assumes that DC reaches the stationary distribution each iteration, which is unrealistic in practice with only a few Langevin steps.\n- With strong convexity of the fidelity term, only convergence to a ball is ensured, not a point. Without strong convexity, an adaptive ADMM penalty schedule is required.\n- The experiments use a pre-trained score model, so the advantages are partly dependent on that backbone.\n- Many baselines are not ADMM-PnP, so the comparison may be apples-to-oranges from an optimization perspective.", "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": "ciPwFpsurt", "reviewer": "Reviewer_JtYg", "summary": "This paper proposes a novel Plug-and-Play (PnP) framework that integrates score-based denoisers into the ADMM optimization algorithm for solving inverse problems. The key contribution is a three-stage denoiser, termed AC-DC, designed to address the manifold mismatch problem: the fact that ADMM iterates (influenced by dual variables) do not lie on the noisy data manifolds on which the score functions were trained. The AC (Auto-Correction) stage adds Gaussian noise to the iterate, while the DC (Directional Correction) stage uses conditional Langevin dynamics to refine the iterate towards the correct noisy manifold before final score-based denoising. The authors provide a comprehensive convergence analysis, showing both fixed-point ball convergence under a constant step size and convergence under an adaptive step size schedule. Experiments across various inverse problems demonstrate improved performance over several baselines.", "questions": "See above", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "- The paper clearly identifies and addresses a significant, underexplored challenge in the PnP literature: the mismatch between the geometry of optimization iterates (especially in primal-dual methods like ADMM) and the manifolds on which score-based denoisers are trained. This is a pertinent and non-trivial issue.\n\n- The proposed AC-DC denoiser is a novel and intuitive solution to the manifold mismatch problem. The idea of proactively \"correcting\" the iterate onto the score function's domain, rather than simply applying the denoiser, is innovative.\n\n- The paper validates the method on a wide range of inverse problems (inpainting, deblurring, super-resolution, phase retrieval) and against a diverse set of modern baselines, showing consistent and often superior performance.", "weaknesses": "1. **Questionable Core Assumption**: A central weakness lies in the justification of the DC step. As outlined in the derivation leading to Eq. (10), the method assumes the residual s^(k) follows a Gaussian prior. This assumption appears to contradict the paper's primary motivation—that the noise in the ADMM iterate is not Gaussian and is off-manifold. If s^(k) can be reasonably modeled as Gaussian, the necessity of the complex DC correction is significantly undermined, as a simpler Gaussian denoiser might suffice. The paper lacks a compelling empirical or theoretical justification for this critical assumption.\n\n2. **Lack of Clarity on Method Variants**: The distinction between the two presented variants, \"Ours-tweedie\" and \"Ours-ode,\" is not clearly explained in the main text. The reader is left to infer the difference from the appendix, which is unsatisfactory. The core methodological description should explicitly state what these variants are.\n\n3. **Insufficient Discussion of Related Work**: The proposed correction mechanism bears a conceptual resemblance to the Onsager correction in Denoising Approximate Message Passing (D-AMP) algorithms [1, 2], which also handles structured iteration noise. Furthermore, the nature of iteration noise in PnP algorithms has been explicitly studied in works like TFPnP [3] (Section 6.2 -- Iteration Noise of PnP Methods). The paper would be significantly strengthened by discussing these connections and clearly delineating how the proposed approach differs from or relates to these ideas.\n\n4. **Inadequate Discussion of Computational Cost**: The AC-DC denoiser, particularly the DC step which involves multiple (J=10) Langevin steps, drastically increases the number of score function evaluations (NFE) per ADMM iteration compared to a standard PnP step. While the appendix includes an NFE analysis, the main text should frankly acknowledge this as a significant limitation and discuss the trade-off between performance and computational efficiency.\n\nGiven these issues, my initial rating is Borderline Reject. However, I would be inclined to raise my score if the authors can:\n- Provide a convincing rationale for the Gaussian assumption in Equation (10),\n- Clarify the differences between \"Ours–tweedie\" and \"Ours–ode,\"\n- Thoroughly discuss relevant prior work (e.g., D-AMP and TFPnP), and \n- Acknowledge and analyze the computational limitations of the AC-DC denoiser.\n\n### References  \n[1] From Denoising to Compressed Sensing, TIT 2016  \n[2] Denoising AMP for MRI Reconstruction: BM3D-AMP-MRI, 2018  \n[3] TFPnP: Tuning-free Plug-and-Play Proximal Algorithms with Applications to Inverse Imaging Problems, JMLR 2022", "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": "AaQdIH9D86", "reviewer": "Reviewer_srVv", "summary": "## Summary\nThis paper proposes an AC-DC denoiser within an ADMM plug-and-play (PnP) framework for inverse problems. The denoiser couples an annealed correction (AC) step with a drift-correction (DC) step inspired by score-based diffusion, and the theory aims to show convergence of ADMM-PnP with this denoiser under high-probability weak nonexpansiveness. Theoretical results include: (i) convergence under strong convexity of the fidelity term via a fixed-point argument; (ii) a high-probability weak nonexpansiveness result under a schedule for σ(k) that decays to zero, leading to convergence with fixed ρ; and (iii) a convergence result without convexity of ℓ by adopting an adaptive-ρ scheme (Chan et al., 2016), under boundedness assumptions on the data domain and the score, and an assumption of bounded gradients of ℓ.", "questions": "Please see the weaknesses part.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "## Strengths\n\n1. The paper tackles an important and timely problem: how to safely and provably integrate score-based denoisers within PnP/ADMM. The AC-DC design is intuitive and practically relevant.\n\n2. The high-probability analysis for weak nonexpansiveness is interesting; to my knowledge, there are relatively few works that attempt to rigorously control the stochasticity induced by score-based denoisers within PnP iterations.\n\n3. The boundedness result under adaptive ρ and a vanishing σ(k) schedule is a useful step toward understanding convergence without convexity.\n\n4. The empirical results suggest the method is competitive across a range of inverse problems.", "weaknesses": "## Major concerns\n\nDespite the nice contributions, I have the following major concerns.\n\n### 1 AC-DC denoiser\n\n> Algorithm 1 injects noise inside the AC-DC denoiser. As implemented, Dσ is a stochastic operator: given the same input, it can return different outputs due to the injected noise (both in the AC step and in the DC sampling/evolution).\nHowever, the convergence analysis treats Dσ as a deterministic mapping z → Dσ(z), i.e., a point-to-point operator. This is a mismatch. It seems that the current deterministic analysis does not rigorously cover the algorithm being evaluated experimentally.\n\n### 2. Theorem 1\n\n> 2.1 The proof strategy follows the fixed-point iteration approach of Ryu et al. This requires the fidelity ℓ to be µ-strongly convex. This is violated in many applications highlighted by the paper itself: deblurring, super-resolution, compressed sensing, MRI, and inpainting (e.g., rank-deficient A or non-strongly-convex penalties).\n\n> 2.2 The parameter condition involving ε and µ may be restrictive. In particular, as ε → 1, the factor (1 + ε − 2ε^2) → 0, making the right-hand side ε/(µ(1 + ε − 2ε^2)) blow up. Then the condition 1/ρ > ε/(µ(1 + ε − 2ε^2)) is practically impossible to satisfy. This contradicts with the experimental setup, where Table 3 indicates ρ ≥ 100 in all cases, which violates the small-step requirement in the theorem.\n\n### 3. Theorem 2\n\n> The assumption “the DC step reaches the stationary distribution for each k” is strong. \n\n### 4. Theorem 3\n\n> 4.1 The analysis follows Chan et al. (2016): boundedness of the denoiser, vanishing noise schedules, and adaptive ρ yield convergence to a fixed point. While this shows stability, it does not characterize the limit point as a solution to any explicit optimization problem. As a result, the algorithm’s fixed point lacks interpretability: it is not known to minimize a well-defined objective nor to satisfy an equilibrium condition like a monotone inclusion. This is a conceptual limitation of the framework that should be acknowledged and discussed more candidly.\n\n> 4.2 The assumption may be strong: The requirement that ||∇ℓ(x)||/√d ≤ R < ∞ for all x is generally false for common inverse problems (deblurring, super-resolution, compressed sensing, MRI, inpainting) unless x is constrained to a compact set. For instance, with ℓ(x) = (1/2σ2)||Ax − y||2, the gradient norm grows with ||x|| unless constrained.\n\n\n## Overall assessment\nThis work addresses an important problem and proposes a practically relevant denoiser within PnP/ADMM, with nontrivial theoretical attempts. However, the current theoretical results rely on assumptions that do not match the algorithm as implemented (notably, treating a stochastic denoiser as deterministic and assuming stationarity per iteration), and on conditions that are violated in key applications and in the reported experiments (strong convexity of ℓ; parameter inequalities incompatible with large ρ; bounded gradient assumptions). I believe the paper would be significantly strengthened by a proper stochastic operator analysis and by reconciling the parameter/schedule assumptions with the experimental setup. \n\nIf the authors address some of the issues mentioned above, I would consider to increase my score.", "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": "2CKYTKBDw3", "reviewer": "Reviewer_zbpZ", "summary": "The paper introduces a new plug-and-play ADMM framework that integrates score-based denoisers derived from pretrained diffusion models. The contribution lies in two main aspects: (1) addressing the  manifold mismatch between the noise-trained score manifolds and the actual ADMM iterates, particularly due to dual-variable effects and (2) the lack of convergence guarantees when using score-based denoisers within ADMM.\n\nThe author propose using a three-stage denoiser called AC–DC (Auto-Correction + Directional Correction + Score Denoising), where in the AC step,  Gaussian noise is added to align iterates with score manifolds, in the DC step  conditional Langevin dynamics is used to refine the direction toward the true data manifold and in the score denoising, pre-trained score model is applied. They provide theoretical results for the cases of convex data-fidelity with weakly contractive denoisers and noncovex case. \n\nExperiments on image inverse problems (super-resolution, inpainting, motion deblurring) show performance gains over methods like DDRM, DiffPIR, and DPS.", "questions": "Could the author comment on the computation complexity added by using the AC-DC denoiser as opposed to simply apply the score-based denoiser with adjusting to the noise manifold?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "The combination of ADMM with a diffusion-based denoiser that explicitly accounts for manifold mismatch  is conceptually fresh. Previous works used diffusion models in proximal steps or as score priors, but rarely in a primal–dual ADMM context with noise manifold adjustment. \n\nThe authors provide theoretical analysis regarding the convergence of the proposed denoiser. \n\nThe experimental results demonstrate that the proposed denoiser is indeed effective in solving inverse problems. \n\nThe approach tackles a real gap between diffusion-based inference and optimization-based inverse problem solvers.", "weaknesses": "Some of the assumptions are strong. Theorem 2 requires stationarity of the inner Langevin DC step at each iteration and smoothness/coercivity of –log p(x); these are hard to ensure in high-dimensional image spaces. Is there anyway authors could test this on the empirical images or a toy problem?", "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)