Domain Adaptation of Mismatched Proximal Denoiser for Plug-and-Play Image Reconstruction Guixian Xu
GXX 422@ STUDENT. BHAM . AC . UK
School of Mathematics, University of Birmingham
Jinglai Li
J . LI .10@ BHAM . AC . UK
School of Mathematics, University of Birmingham
Junqi Tang
J . TANG .2@ BHAM . AC . UK
arXiv:2607.14894v1 [eess.IV] 16 Jul 2026
School of Mathematics, University of Birmingham
Abstract Plug-and-play proximal gradient descent (PnP-PGD) enables flexible image reconstruction by using denoisers as implicit priors. In practice, these denoisers are often deployed outside their training domains. Existing analyses establish convergence under structural assumptions on the deployed denoiser, such as requiring it to be a proximal map or a contraction. However, they do not measure how domain mismatch affects convergence of PnP-PGD. We define this effect as proximal mismatch: the discrepancy between a deployed b and a target-domain reference map D⋆ = prox associated with the underlying regularizer denoiser D R⋆ R⋆ . Under this mismatch, each denoising update becomes an inexact proximal step for the target objective. We further derive a stationarity bound that decays at a rate of O(1/K), with an additive term proportional to the average squared proximal mismatch. This result motivates adaptation via proximal matching rather than MSE-based adaptation alone. We study this approach with two established denoiser families: learned proximal networks and gradient-step denoisers. Experiments on Gaussian deblurring and super-resolution under substantial domain shift show that proximal matching adaptation improves reconstruction quality significantly over MSE-based adaptation, yielding the largest numerical gains in the few-shot regime.
1. Introduction Imaging inverse problems recover an unknown image from noisy measurements by combining a forward physical model with a prior. Plug-and-play (PnP) methods replace the regularization step of an iterative optimization algorithm with a trained denoiser (Venkatakrishnan et al., 2013; Chan et al., 2017; Kamilov et al., 2023). This design separates the prior from the forward model and allows the same denoiser to be used with different imaging operators. However, a generic denoiser may not be the proximal map of any regularizer, so the resulting iterations may not minimize an explicit objective (Buzzard et al., 2018;
1
Kamilov et al., 2023). Existing convergence analyses have therefore studied PnP as a fixed-point iteration under conditions such as nonexpansiveness and contractivity (Chan et al., 2017; Ryu et al., 2019). More recent work has designed denoisers with proximal structure, including convolutional proximal networks, gradient-step denoisers, and learned proximal networks (Hertrich et al., 2021; Hurault et al., 2022a,b; Fang et al., 2024). Under suitable conditions, these denoisers allow PnP-PGD to be analyzed as proximal gradient descent on an explicit objective and provide first-order convergence guarantees. Despite this progress, PnP remains sensitive to prior mismatch. A denoiser trained on one image distribution may be deployed on another, creating a mismatch between the learned and target priors. This mismatch is difficult to correct in few-shot scenarios with limited target data. Even if the deployed denoiser satisfies the conditions required for convergence, the regularizer it represents may not match the target domain. Existing work has analyzed prior mismatch in SD-RED and PnP-ADMM and has explored test-time adaptation for PnP (Shoushtari et al., 2022; Chandler et al., 2023; Shoushtari et al., 2024). Related studies show that denoiser inputs along PnP trajectories can differ from standard Gaussian denoising samples (Vo et al., 2024), and that proximal or MAP behavior differs from MMSE denoising (Fermanian et al., 2023; Vert et al., 2026). Thus, denoising MSE alone may not target the operator behavior required by PnP reconstruction. In the structured proximal setting, this behavior is represented by a target-domain proximal map. What remains unclear is how replacing this map with a mismatched denoiser affects convergence of the target objective in PnP-PGD and how this effect should guide adaptation. This issue matters because MSE-based denoiser training targets a conditional mean but does not directly enforce agreement with a target proximal map. Therefore, better denoising performance does not necessarily reduce the proximal error encountered during PnP-PGD. b with a target-domain We analyze PnP-PGD under domain shift by comparing the deployed denoiser D reference proximal map D⋆ = proxR⋆ . This reference map corresponds to the target regularizer R⋆ and b turns each denoising update into an inexact defines the target objective F⋆ = ηf + R⋆ . Replacing D⋆ with D proximal step for F⋆ . We define proximal mismatch as the discrepancy between their outputs at the query points generated by PnP-PGD. We then derive a first-order stationarity bound for PnP-PGD. This bound decays at a rate of O(1/K) and includes an additive term proportional to the average squared proximal mismatch. This result links domain shift directly to convergence of the target objective. Table 1 positions our analysis relative to the most closely related work on inexact proximal methods, prior mismatch, and structured PnP denoisers. This bound suggests a direct adaptation strategy. Instead of fine-tuning the denoiser with MSE alone, we use proximal matching to align it with the target-domain proximal map (Fang et al., 2024). We study this strategy with two structured denoiser families: learned proximal networks and gradient-step denoisers. This allows us to test whether the same adaptation principle works across two different ways of realizing proximal maps. We evaluate our method on Gaussian deblurring and super-resolution under substantial domain shifts. Proximal-matching adaptation reduces the proximal mismatch along the PnP-PGD trajectory and improves reconstruction quality over MSE-based adaptation. In severe mismatch settings, it improves reconstruction PSNR by more than 10 dB, with the largest gains in the few-shot regime. Our contributions are threefold: • Proximal-mismatch analysis. We formulate domain shift in PnP-PGD as proximal mismatch. We show that this mismatch induces an inexact proximal step and derive an O(1/K) stationarity bound with an additive term proportional to the average squared proximal mismatch.
2
Work
Algorithm
Prior / denoiser model
Mismatch / error model
Assumptions
PGD RED
Explicit proximal operator MMSE denoiser
Inexact proximal steps Prior mismatch
ϵ-optimal BD, BI
PnP-PGD PnP-PGD PnP-ADMM PnP-PGD PnP-PGD
MMSE denoiser Proximal denoiser* MMSE denoiser Proximal denoiser Proximal denoiser
Prior match Prior match Prior mismatch Prior match Proximal mismatch
TD TD BD, BI TD ER
Classical / RED-type analyses Schmidt et al. (2011) Shoushtari et al. (2022) Plug-and-Play analyses Xu et al. (2020) Hurault et al. (2022b) Shoushtari et al. (2024) Fang et al. (2024) Ours
Table 1: Theoretical context for the convergence analysis developed in this paper. The table compares representative proximal-gradient, RED, and PnP analyses in terms of algorithm, denoiser model, mismatch/error model, and assumptions. TD = matched target denoiser used at inference; BD = bounded denoiser; BI = bounded iterates; ER = target-envelope regularity. ’*’ indicates a proximal interpretation that holds under additional conditions. • Proximal-matching adaptation. Guided by this bound, we propose a few-shot adaptation method based on proximal matching rather than MSE-based adaptation. We apply it to learned proximal networks and gradient-step denoisers. • Experimental validation. We evaluate the proposed method on Gaussian deblurring and superresolution under substantial domain shifts. Proximal matching adaptation reduces the proximal mismatch along the PnP-PGD trajectory and improves reconstruction PSNR over MSE-based adaptation, with gains exceeding 10 dB in severe settings and the largest gains in the few-shot regime.
2. Background and Problem Setup Inverse Problems.
We consider the recovery of an unknown signal x ∈ Rn from a noisy measurement y = Ax + e,
where A ∈ Rm×n is a known forward operator and e ∈ Rm denotes measurement noise. The inverse problem is often ill-posed: a solution may not exist, uniqueness may fail, or small perturbations in the measurements may cause large changes in the recovered signal (Engl et al., 1996; Hansen, 2010). Regularization addresses this instability by adding prior information about the unknown signal. Given a regularizer R, we consider minn F (x) := ηf (x) + R(x), (1) x∈R
where f measures consistency with the observations, R favors plausible solutions, and η > 0 controls the relative weight of the data-fidelity term.
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Plug-and-Play Denoisers A common way to solve (1) is to use first-order proximal splitting algorithms (Beck and Teboulle, 2009; Parikh and Boyd, 2014). It handles the data-fidelity term by a gradient step and the regularizer by its proximal map. We define 1 2 := proxR (z) arg minn ∥x − z∥ + R(x) , (2) x∈R 2 where the proximal parameter is absorbed into R. With the normalization in (1), proximal gradient descent takes the form z k+1 = xk − η∇f (xk ), xk+1 = proxR (z k+1 ). (3) This splitting viewpoint naturally motivates Plug-and-Play (PnP) methods (Venkatakrishnan et al., 2013; Chan et al., 2017; Kamilov et al., 2023), which replace the proximal operator of R with an image denoiser D: z k+1 = xk − η∇f (xk ), xk+1 = D(z k+1 ). (4) The denoiser is commonly trained to remove additive white Gaussian noise and is then used as an implicit image prior (Zhang et al., 2021; Kamilov et al., 2023). This design allows PnP-PGD to use learned priors without specifying R explicitly. However, a generic denoiser need not be the proximal map of any regularizer, so the resulting iteration may not minimize an explicit objective. In this work, we focus on denoisers that admit a proximal representation (Hertrich et al., 2021; Hurault et al., 2022a,b; Fang et al., 2024). This setting allows PnP-PGD to be analyzed as proximal gradient descent on an explicit objective. 2.1 Structured Proximal Denoisers A denoiser D has a proximal representation if there exists a regularizer R such that D = proxR . This structure allows PnP-PGD to be related to an explicit optimization objective. We represent the target domain by a regularizer R⋆ and define its reference proximal map as D⋆ = proxR⋆ .
(5)
F⋆ (x) = ηf (x) + R⋆ (x).
(6)
The corresponding target objective is
Using D⋆ in (4) recovers proximal gradient descent on F⋆ under the normalization introduced above. b This denoiser may be trained on a source domain or In practice, PnP-PGD uses a deployed denoiser D. adapted using limited target-domain data. Its outputs may therefore differ from those of D⋆ at the query points generated by PnP-PGD. We study how this discrepancy affects stationarity of F⋆ . This notation separates the target model from the deployed operator. The map D⋆ defines the target regularizer and objective, b is the operator used during reconstruction. The next two paragraphs describe two denoiser families while D that realize proximal maps in different ways: learned proximal networks and gradient-step denoisers (Hurault et al., 2022a,b; Fang et al., 2024). Learned Proximal Networks. Learned Proximal Networks (LPNs) are designed so that the learned denoising map is an exact proximal operator of an associated learned regularizer (Fang et al., 2024). More specifically, they parameterize the denoiser through the gradient of a strongly convex potential, which yields
4
a globally defined proximal map and an induced, generally nonconvex, regularization function. In our setting, this architecture-level proximality provides a direct realization of the target reference D⋆ = proxR⋆ . Consequently, the target objective F⋆ = ηf + R⋆ is not merely implicit in the iteration but is explicitly tied to the learned proximal map. Gradient-Step Proximal Denoisers. take the form
A second family is given by Gradient-Step (GS) denoisers, which Dθ = Id −∇gθ ,
(7)
where gθ is a scalar-valued neural potential (Cohen et al., 2021; Hurault et al., 2022a). This parameterization imposes a conservative structure on the denoiser, since the residual Id −Dθ is represented as the gradient of a potential. Under suitable smoothness and contractivity conditions, such maps can be interpreted as proximal operators of induced regularizers (Hurault et al., 2022b). GS denoisers therefore provide another route to a well-defined proximal reference, but with a different guarantee mechanism from LPNs: LPNs build proximality directly into the architecture, whereas GS denoisers obtain proximal structure through analytic conditions on the learned potential. Together, these two examples define the proximal-structured setting considered in this paper. In both cases, b This one can distinguish between the ideal target proximal map D⋆ and the deployed inference denoiser D. distinction is the basis for the subsequent analysis of how denoiser mismatch, or prior shift, propagates into stationarity errors for the target objective F⋆ .
3. Theory Our analysis consists of three main steps. First, we show how the mismatch between the deployed and target proximal denoisers affects the target proximal update. Second, we prove that the target objective still decreases, up to an error caused by this mismatch. Third, we show that the target proximal step approximately satisfies its optimality condition. Putting these results together, we obtain a bound on the average stationarity gap, where the remaining error depends on the mismatch gap. Finally, we show that the assumptions used in our analysis hold for Learned Proximal Networks and Gradient-Step denoisers. 3.1 Target Objective and Basic Assumptions All stationarity statements are made with respect to the target objective (6) defined by the target proximal reference D⋆ = proxR⋆ . The map D⋆ specifies the target prior and the objective whose stationary points are b determines the iterates generated by PnP-PGD. of interest, whereas the deployed denoiser D We measure target stationarity through ∥∇F⋆ (x)∥. Accordingly, R⋆ is assumed to be differentiable at the deployed iterates. The additional regularity required for the target proximal-subproblem is stated below and verified for LPN and GS target references in Section 3.4. Assumption 3.1 (Smooth data fidelity) The data-fidelity term f : Rn → R is differentiable and has Lf Lipschitz continuous gradient. Assumption 3.2 (Lower bounded target objective) The target objective F⋆ is bounded below: F⋆inf := inf F⋆ (x) > −∞. x
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3.2 Proximal Mismatch as Inexact Proximal Step b For k ≥ 1, define the query Let {xk }k≥0 be the PnP-PGD sequence generated by the deployed denoiser D. point z k = xk−1 − η∇f (xk−1 ). (8) The deployed update is b k ). xk = D(z
(9)
At the same query point, the target proximal response is D⋆ (z k ). We define the proximal error as b k ) − D⋆ (z k ), ξ k := D(z
(10)
b k ) − D⋆ (z k ) . dk := ∥ξk ∥2 = D(z 2
(11)
and the proximal mismatch as
When a deterministic upper bound is available, we write dk ≤ ∆k . The deployed update can now be written as xk = D⋆ (z k ) + ξ k .
(12)
b in place of D⋆ gives an inexact proximal step for the target objective. Thus, using D To connect dk to standard measures of proximal inexactness, define the target proximal subproblem 1 Hk (u) := ∥u − z k ∥22 + R⋆ (u). 2
(13)
Since D⋆ (z k ) ∈ arg min Hk (u), u
we define the proximal subproblem error as b k ) − Hk D⋆ (z k ) ≥ 0. ϵk := Hk D(z
(14)
Lemma 3.1 (Proximal mismatch controls proximal subproblem error) Fix k ≥ 1 and let Hk be defined b k ). by (13). Assume that Hk is differentiable and LH -smooth on the line segment between D⋆ (z k ) and D(z Then LH b LH 2 2 ϵk ≤ D(z k ) − D⋆ (z k ) 2 = d . (15) 2 2 k Consequently, if dk ≤ ∆k , then LH 2 ∆ . (16) ϵk ≤ 2 k Proof. See Appendix A.1.
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3.3 Convergence Analysis The next assumption summarizes the regularity needed to convert proximal inexactness into a stationarity bound. Assumption 3.3 (Proximal subproblem regularity) For every visited query point z k , the target proximal subproblem Hk has the unique minimizer D⋆ (z k ). Moreover, Hk satisfies Lemma 3.1 with a constant LH b k ), independent of k, and at the deployed iterate xk = D(z ∥∇Hk (xk )∥2 ≤ 2LH Hk (xk ) − Hk (D⋆ (z k )) . (17) In the LPN and GS realizations below, this condition follows because the proximal subproblem can be written, up to constants, as a smooth convex potential minus a linear term. In particular, ∇Hk (u) = D−1 ⋆ (u) − z k , and the Lipschitz regularity of D−1 ⋆ provides a uniform smoothness constant. Lemma 3.2 (Descent with inexact target-proximal steps) Under Assumption 3.1, the deployed PnP-PGD iterates satisfy 1 − ηLf F⋆ (xk+1 ) ≤ F⋆ (xk ) − ∥xk+1 − xk ∥2 + ϵk+1 . (18) 2 Proof. See Appendix A.2. Lemma 3.3 (Stationarity residual control) Under Assumptions 3.3, for each k ≥ 0 there exists a residual vector r k+1 such that ∇R⋆ (xk+1 ) = z k+1 − xk+1 − r k+1 ,
∥r k+1 ∥2 ≤ 2LH ϵk+1 .
(19)
Proof. See Appendix A.3. Theorem 3.1 (Convergence under proximal mismatch) Suppose that Assumptions 3.1–3.3 hold and that ηLf < 1. Define ! LH 16 16 inf C0 := F⋆ (x0 ) − F⋆ , C1 := + 4LH . (20) 1 − ηLf 1 − ηLf 2 Then, for all t ≥ 1, t
t
1X C0 1X 2 ∥∇F⋆ (xk )∥2 ≤ + C1 dk . t t t k=1
k=1
Consequently, t
C0 1X 2 + C1 dk . 1≤k≤t t t k=1 P 2 If dk ≤ ∆k , the same bounds hold with dk replaced by ∆k . If ∞ k=1 dk < ∞, then min ∥∇F⋆ (xk )∥2 ≤
∥∇F⋆ (xk )∥ → 0. In particular, this holds whenever d2k = O(k −1−ν ) for some ν > 0. 7
(21)
Proof. See Appendix A.4. b = D⋆ , then dk = 0 for all k, and Theorem 3.1 Corollary 3.1 (Matched target proximal denoiser) If D gives t 1X 16 F⋆ (x0 ) − F⋆inf . (22) ∥∇F⋆ (xk )∥2 ≤ t (1 − ηLf )t k=1
Proof. See Appendix A.5. 3.4 Structured Proximal-Denoiser Realizations The preceding result applies whenever the target prior admits a proximal reference satisfying the regularity conditions above. We instantiate this abstract setting with two structured proximal denoiser families: Learned Proximal Networks, which provide architecture-level proximal maps, and Gradient-Step proximal denoisers, which provide proximal maps under contractivity conditions. Detailed proximal characterizations and derivations are provided in Appendix B. 3.4.1 L EARNED P ROXIMAL N ETWORKS Following Fang et al. (2024), an LPN is parameterized as Dθ (x) = ∇Ψθ (x),
Ψθ (x) = ψθ (x) +
α ∥x∥2 , 2
α ∈ (0, 1),
(23)
where ψθ is an input-convex neural network with C 2 activations. Such a network is an exact proximal map Dθ = proxRθ , and its inverse is 1/α-Lipschitz; see Proposition B.1 in the appendix. For the target denoiser D⋆ = Dθ⋆ , the associated proximal subproblem satisfies, up to additive constants, Hk (u) = Ψ⋆θ⋆ (u) − ⟨z k , u⟩. Consequently, Hk is convex and 1/α-smooth. Therefore, Assumption 3.3 holds with LH =
1 , α
and Lemma 3.1 yields ϵk ≤
1 b 2 Dθ (z k ) − Dθ⋆ (z k ) . 2α
(24)
Hence Theorem 3.1 applies with b θ (z k ) − Dθ⋆ (z k ) . dk = D 3.4.2 G RADIENT-S TEP P ROXIMAL D ENOISERS GS denoisers take the form 1 hσ (x) = ∥x∥2 − gσ (x). 2
Dσ = ∇hσ = Id −∇gσ ,
(25)
Suppose that ∇gσ,⋆ is L-Lipschitz with L < 1. Then the target GS denoiser Dσ,⋆ is a proximal map on its image, and its inverse is 1/(1 − L)-Lipschitz; see Proposition B.2 in the appendix. 8
Assume additionally that b σ (z k ) ∈ Im(Dσ,⋆ ) D at every visited query point. On this set, the associated proximal subproblem satisfies, up to additive constants, Hk (u) = h⋆σ,⋆ (u) − ⟨z k , u⟩. Thus Hk is convex and 1/(1 − L)-smooth, so that LH =
1 . 1−L
Accordingly, ϵk ≤
1 b σ (z k ) − Dσ,⋆ (z k ) 2 , D 2(1 − L)
(26)
and Theorem 3.1 applies with b σ (z k ) − Dσ,⋆ (z k ) . dk = D LPNs and GS denoisers therefore instantiate the same proximal-mismatch theorem through different mechanisms: LPNs yield globally defined proximal maps by construction, whereas GS denoisers yield proximal maps on the image of the target denoiser under the condition L < 1.
4. Proximal Matching Adaptation Theorem 3.1 shows that target stationarity depends on the average deployed-to-target proximal discrepancy along the PnP-PGD trajectory. This result motivates adapting the deployed denoiser toward the target proximal map. In practice, direct evaluations of the target proximal map are unavailable during few-shot adaptation. We instead use a clean target-domain set n
adapt n Star = {xi }i=1
and generate Gaussian-corrupted inputs y i,j = xi + σadapt δ i,j ,
δ i,j ∼ N (0, I),
j = 1, . . . , M.
(27)
We first compare the population targets of squared-error regression and proximal matching. We then apply proximal matching to LPN and GS denoisers. 4.1 Population Targets of MSE and Proximal Matching Let X ∼ ptar ,
δ ∼ N (0, I).
Y = X + σadapt δ,
Then ∥y − x∥22 ptar (x | y) ∝ exp − 2 2σadapt
9
! ptar (x).
For Rtar (x) = − log ptar (x), the MAP denoiser is 1 2 2 DMAP (y) ∈ arg min ∥x − y∥2 + σadapt Rtar (x) = proxσ2 Rtar (y). adapt x 2 For u, v ∈ Rd , define ∥u − v∥22 ργ (u, v) = 1 − exp − 2γ 2
(28)
! ,
γ > 0.
(29)
Proposition 4.1 (Population targets) Let D : Rd → Rd range over all measurable denoisers, and assume that E∥X∥22 < ∞. The population MSE risk h i RMSE (D) = E ∥D(Y ) − X∥22 (30) is minimized by D⋆MSE (y) = E[X | Y = y].
(31)
RPM,γ (D) = E ργ (D(Y ), X)
(32)
The population PM risk has conditional minimizers D⋆PM,γ (y) ∈ arg max E exp z
∥z − X∥22 − 2γ 2
!
Y = y .
(33)
Thus, PM targets the mode of a Gaussian-smoothed posterior. Under the consistency conditions of Fang et al. (2024), if the posterior mode is unique, then D⋆PM,γ (y) −→ DMAP (y)
as γ ↓ 0.
(34)
Proof See Appendix A.6. MSE targets the posterior mean, whereas PM approaches the MAP/proximal denoiser. This motivates using PM to reduce proximal mismatch. 4.2 LPN Adaptation Let Dθs be a source-trained LPN. We fine-tune the same proximal parameterization, Dθ = ∇Ψθ , on the target adaptation set. Clean-MSE adaptation.
The MSE baseline minimizes LMSE (θ) =
1 nadapt M
nadapt M X X
Dθ (y i,j ) − xi
i=1 j=1
2 2
.
(35)
Starting from θs , we obtain θMSE ∈ arg min LMSE (θ). θ∈ΘLPN
10
(36)
Proximal-matching adaptation.
PM minimizes
LPM (θ; γ) =
1
nadapt M X X
nadapt M
i=1 j=1
ργ Dθ (y i,j ), xi .
(37)
Starting from θs , we obtain θPM ∈ arg min LPM (θ; γ).
(38)
θ∈ΘLPN
The bandwidth γ may be annealed during fine-tuning. Both objectives preserve the LPN parameterization but have different population targets. 4.3 Gradient-Step Proximal Adaptation A GS denoiser has the form DGS θ (u) = u − ∇gθ (u),
(39)
where gθ : Rd → R is differentiable. Its proximal interpretation requires ∇gθ to be contractive. Clean-MSE adaptation.
The GS MSE baseline is LGS-MSE (θ) =
Gradient-field proximal matching.
1
nadapt M X X
nadapt M
i=1 j=1
DGS θ (y i,j ) − xi
2 2
.
(40)
Because
h i ) − x = − ∇g (y ) − (y − x ) DGS (y i i , θ i,j i,j i,j θ output matching is equivalent to matching the gradient field to the corruption residual. We define LGS-PM (θ; γ) =
Contractivity penalty. 1, define
1
nadapt M X X
nadapt M
i=1 j=1
ργ ∇gθ (y i,j ), y i,j − xi .
(41)
b Let L(∇g θ ) estimate the Lipschitz constant on the adaptation inputs. For Lmax < h i2 b Rcon (θ) = L(∇g . θ ) − Lmax
(42)
LGS-AdaPM (θ) = LGS-PM (θ; γ) + λcon Rcon (θ).
(43)
+
The final objective is The PM term adapts the gradient field. The penalty promotes the contractivity condition required by the GS proximal interpretation.
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Figure 1: Representative images from the source and target domains. BreCaHAD contains histopathology patches, whereas CelebA contains aligned facial images, producing a substantial source-target shift in both semantic content and visual statistics.
5. Numerical Experiments Our experiments address three questions. First, how much does deploying a source-trained denoiser degrade PnP-PGD reconstruction on the target domain? Second, in the few-shot regime, does proximal-matching adaptation outperform MSE-based adaptation? Third, are reconstruction gains accompanied by lower proximal mismatch along the PnP-PGD trajectory? We study these questions under a substantial BreCaHAD (Liu et al., 2015)-to-CelebA (Aksac et al., 2019) domain shift for Gaussian deblurring and single-image super-resolution. We consider two structured denoiser families: LPNs and GS denoisers. For each family, we compare the deployed denoiser with a target-trained proximal reference from the same family and evaluate both reconstruction performance and on-trajectory proximal mismatch. 5.1 Experimental Setup Data and domain shift. We train the source denoisers on BreCaHAD and perform few-shot adaptation and reconstruction on CelebA (Aksac et al., 2019; Liu et al., 2015). BreCaHAD contains breast histopathology images with cellular morphology and stain-dependent appearance, whereas CelebA contains aligned face images. This source-to-target shift changes both semantic content and low-level image statistics. For each inverse problem, we evaluate all methods on the same 100 images from the CelebA test set. The target-domain adaptation set is disjoint from the test set and contains nadapt ∈ {1, 5, 25, 50} images. We report the mean and standard deviation of the reconstruction metrics across the 100 test images. Figure 1 shows representative images from the source and target domains. 12
Gaussian deblurring. to
We generate blurred observations from RGB images x ∈ [0, 1]3×128×128 according y = Adb x + e,
Adb = Hg ,
(44)
where Hg applies a 5 × 5 Gaussian kernel with standard deviation 1.0 independently to each color channel, and e ∼ N (0, 0.022 I). We use the data-fidelity term 1 fdb (z) = ∥Adb z − y∥22 . 2 2 For the implemented operator, Ldb = ∥Adb ∥2 = 1. We run PnP-PGD for 40 iterations with ηdb = 0.95 and initialize x0 = A⊤ db y. Thus, ηdb Ldb = 0.95 < 1. Single-image super-resolution.
For ×4 super-resolution, we use y = Asr x,
Asr = S4 Hσsr ,
(45)
where Hσsr applies a normalized 25× 25 Gaussian anti-aliasing kernel with σsr = 1.6 independently to each color channel, and S4 downsamples the result by a factor of four in each spatial dimension. This operator maps an RGB 128 × 128 image to an RGB 32 × 32 measurement. We add no measurement noise. The data-fidelity term is 1 fsr (z) = ∥Asr z − y∥22 . 2 We initialize PnP-PGD by bicubic upsampling followed by pixel-shift correction. For the implemented operator, Lsr = ∥Asr ∥22 ≈ 0.063. We run 40 iterations with ηsr = 15, giving ηsr Lsr ≈ 0.945 < 1. Denoiser families and adaptation baselines. We evaluate two structured denoiser families. LPNs enforce proximal structure by construction, whereas GS denoisers admit a proximal interpretation under contractivity conditions on the learned gradient field. For each family, we consider four variants: a source denoiser trained on BreCaHAD, a target-reference denoiser trained on CelebA, an MSE-adapted source denoiser, and a proximal-matching-adapted source denoiser. The target-reference model serves as the proximal reference when evaluating on-trajectory mismatch. For GS, proximal-matching adaptation also includes the contractivity regularizer defined in Section 4.3; we denote this model by GS-AdaPM. Within each family and adaptation budget, the MSE and proximalmatching variants use the same target images, corruption process, source initialization, and number of adaptation epochs. LPN-MSE and LPN-PM are adapted for 200 epochs, whereas GS-MSE and GS-AdaPM are adapted for 20 epochs. For each inverse problem, all denoisers are evaluated with the same PnP-PGD reconstruction settings. Evaluation metrics. We report PSNR and SSIM for reconstruction quality. To measure proximal mismatch, we also compute a family-matched trajectory gap. For a candidate denoiser D and the targetreference denoiser Dθ⋆ , we evaluate both denoisers on the same PnP query zk at each iteration and compute gap(zk ; D, Dθ⋆ ) =
∥D(zk ) − Dθ⋆ (zk )∥2 . max(∥Dθ⋆ (zk )∥2 , 10−12 )
The reported gap is averaged over PnP iterations and test images. This metric is always interpreted within the same denoiser family, i.e., LPN gaps compare LPN denoisers to the LPN target reference and GS gaps compare GS denoisers to the GS target reference. For GS models, we also report empirical local Jacobian diagnostics for ∇gθ . 13
5.2 Main Deblurring Results Table 2 and Figure 2 report the Gaussian deblurring results under the BreCaHAD-to-CelebA domain shift. The source-trained models show a large loss in reconstruction quality. LPN-source and GS-source achieve 11.52 and 19.75 dB, respectively, compared with 31.40 and 31.75 dB for the corresponding target-trained references. The source models also exhibit nonzero relative trajectory mismatch, showing that the reconstruction loss is accompanied by disagreement with the target proximal responses.
(b) LPN gap Mean trajectory gap
(a) LPN PSNR PSNR (dB)
30 20 10 1 5
25 Adaptation size
50
0.75 0.50 0.25 0.00
1 5
30 25 20 15 1 5
25 Adaptation size
Target ref.
LPN-MSE
50
(d) GS gap Mean trajectory gap
PSNR (dB)
(c) GS PSNR
25 Adaptation size
50
10 1 10 2 10 3 10 4
LPN-PM
1 5
25 Adaptation size
pure GS-MSE
50
GS-AdaPM
Figure 2: Gaussian deblurring performance under the BreCaHAD-to-CelebA domain shift. PSNR and relative trajectory mismatch are shown as functions of the number of target-domain adaptation images. Proximal-matching adaptation yields the largest PSNR gains in the low-data regime and reduces the mismatch with the corresponding target-trained reference. Higher PSNR and lower mismatch are better. For LPNs, proximal matching achieves higher PSNR and lower relative trajectory mismatch than MSE adaptation at every adaptation budget. The PSNR gains over LPN-MSE are 2.86, 4.34, 1.69, and 1.06 dB for nadapt ∈ {1, 5, 25, 50}, respectively. With one target image, MSE adaptation performs below the source model, whereas LPN-PM improves the source result from 11.52 to 12.72 dB. The largest advantage occurs with five target images and decreases as more adaptation data become available. This trend shows that proximal matching is most useful in the few-shot regime.
14
Table 2: Gaussian deblurring under severe BreCaHAD-to-CelebA proximal mismatch. Values are mean ± standard deviation over 100 test images. SSIM and trajectory gap (Gap) are reported in units of 10−2 . b max denotes the The gap is computed against the target-trained reference within each denoiser family. L empirical local Jacobian estimate for GS denoisers. Method
nadapt
PSNR ↑
SSIM (10−2 ) ↑
Gap (10−2 ) ↓
b max ↓ L
LPN proximal denoiser LPN-source LPN-reference
– –
11.52 ± 2.36 31.40 ± 2.89
37.3 ± 18.9 89.7 ± 4.2
82.90 ± 43.84 0
– –
LPN-MSE LPN-PM
1
9.86 ± 1.61 12.72 ± 2.43
32.8 ± 14.3 50.5 ± 14.6
67.01 ± 27.15 43.77 ± 21.49
– –
LPN-MSE LPN-PM
5
23.52 ± 3.09 27.86 ± 2.66
77.5 ± 6.9 84.8 ± 4.8
6.51 ± 2.99 3.12 ± 0.98
– –
LPN-MSE LPN-PM
25
27.89 ± 3.36 29.58 ± 2.94
86.0 ± 4.4 87.8 ± 4.1
3.08 ± 1.56 2.11 ± 1.03
– –
LPN-MSE LPN-PM
50
29.25 ± 3.01 30.31 ± 2.70
87.4 ± 4.1 88.5 ± 4.1
2.18 ± 1.07 1.69 ± 0.78
– –
GS proximal denoiser GS-source GS-reference
– –
19.75 ± 1.09 31.75 ± 2.89
49.9 ± 8.6 91.4 ± 3.4
15.50 ± 4.73 0
1.039 0.927
GS-MSE GS-AdaPM
1
12.29 ± 0.94 24.98 ± 1.14
16.8 ± 4.1 69.3 ± 3.6
48.68 ± 11.28 7.70 ± 1.49
6.567 1.446
GS-MSE GS-AdaPM
5
15.17 ± 0.74 25.17 ± 1.11
25.0 ± 4.5 70.0 ± 3.5
32.24 ± 7.31 7.41 ± 1.31
5.077 1.005
GS-MSE GS-AdaPM
25
28.18 ± 1.61 30.28 ± 2.28
81.0 ± 3.4 88.2 ± 3.0
4.08 ± 0.85 2.44 ± 0.54
0.989 0.985
GS-MSE GS-AdaPM
50
28.15 ± 1.65 30.45 ± 2.33
81.1 ± 3.4 88.7 ± 3.0
4.17 ± 0.84 2.31 ± 0.52
0.992 0.986
The GS results show a stronger difference between the two adaptation objectives in the low-data regime. With one and five target images, GS-MSE drops below the source-model PSNR, while GS-AdaPM reaches 24.98 and 25.17 dB. These results correspond to gains of 12.69 and 10.00 dB over GS-MSE. With 25 and 50 images, GS-AdaPM improves PSNR by 2.10 and 2.30 dB, respectively. It also produces lower relative trajectory mismatch at every adaptation budget. For one-shot and five-shot adaptation, the empirical Jacobian estimates of GS-AdaPM are much smaller than those of GS-MSE. At larger adaptation budgets, both methods yield estimates close to one. These results indicate that the contractivity-regularized proximal-matching objective better preserves the gradient-step structure when target data are limited. The reconstructions in Figure 3 follow the quantitative results. The source-trained denoisers produce strong artifacts after deployment on CelebA. Both adaptation objectives reduce these artifacts, but the proximalmatching variants recover sharper facial contours and retain more local detail than the corresponding MSE
15
Figure 3: Representative Gaussian deblurring results for nadapt = 50 under the BreCaHAD-to-CelebA domain shift. The source-trained denoisers produce visible domain-shift artifacts. LPN-PM and GS-AdaPM recover sharper facial contours and fewer local distortions than their MSE-adapted counterparts. The boxes mark the enlarged regions shown below each reconstruction.
16
baselines. Across both denoiser families, lower relative trajectory mismatch is associated with higher deblurring PSNR. 5.3 Main Super-Resolution Results Table 3 and Figure 4 report the ×4 super-resolution results under the BreCaHAD-to-CelebA domain shift. The source-trained models show a large loss in reconstruction quality. LPN-source achieves 9.94 dB, compared with 27.67 dB for the target-trained LPN reference. GS-source achieves 18.08 dB, compared with 27.67 dB for the target-trained GS reference. Both source models also exhibit nonzero relative trajectory mismatch, showing that the reconstruction loss is accompanied by disagreement with the corresponding target proximal responses.
(a) LPN PSNR Mean trajectory gap
30 PSNR (dB)
25 20 15 10 1
5
25
50
1.0 0.8 0.6 0.4 0.2 0.0
(b) LPN gap
1
5
25 20 15 1
5 25 Adaptation size MSE
PM
50
(d) GS gap Mean trajectory gap
PSNR (dB)
(c) GS PSNR
25
50
10 1 10 3 10 5 1
pure GS-MSE
5 25 Adaptation size GS-AdaPM
50
Target ref.
Figure 4: ×4 super-resolution performance under the BreCaHAD-to-CelebA domain shift. PSNR and relative trajectory mismatch are shown as functions of the number of target-domain adaptation images. Proximal-matching adaptation yields the largest PSNR gains in the low-data regime and reduces the mismatch with the corresponding target-trained reference. Higher PSNR and lower mismatch are better. For LPNs, proximal matching achieves higher PSNR, higher SSIM, and lower relative trajectory mismatch than MSE adaptation at every adaptation budget. The PSNR gains over LPN-MSE are 2.49, 3.53, 1.56, and 0.45 dB for nadapt ∈ {1, 5, 25, 50}, respectively. With one target image, LPN-MSE performs below the source model, whereas LPN-PM improves the source PSNR from 9.94 to 11.97 dB. The largest gain occurs
17
Table 3: Super-resolution results under severe BreCaHAD-to-CelebA proximal mismatch. Values are mean ± standard deviation over 100 test images. SSIM and trajectory gap (Gap) are reported in units of 10−2 . b max denotes the The gap is computed against the target-trained reference within each denoiser family. L empirical local Jacobian estimate for GS denoisers. Method
nadapt
PSNR ↑
SSIM (10−2 ) ↑
Gap (10−2 ) ↓
b max ↓ L
LPN proximal denoiser LPN-source LPN-reference
– –
9.94 ± 1.76 27.67 ± 2.34
23.7 ± 14.3 81.2 ± 5.8
89.61 ± 39.40 0
– –
LPN-MSE LPN-PM
1
9.48 ± 1.56 11.97 ± 2.56
24.8 ± 13.3 38.5 ± 15.4
72.28 ± 28.46 49.53 ± 23.08
– –
LPN-MSE LPN-PM
5
21.54 ± 2.74 25.07 ± 2.10
65.1 ± 8.4 73.9 ± 6.0
6.79 ± 3.52 3.12 ± 1.12
– –
LPN-MSE LPN-PM
25
25.31 ± 2.56 26.87 ± 2.20
75.2 ± 6.2 78.8 ± 5.5
2.88 ± 1.75 1.93 ± 0.98
– –
LPN-MSE LPN-PM
50
26.73 ± 2.13 27.18 ± 2.14
78.0 ± 5.5 79.4 ± 5.5
1.92 ± 1.01 1.58 ± 0.77
– –
GS proximal denoiser GS-source GS-reference
– –
18.08 ± 1.10 27.67 ± 2.35
37.9 ± 8.2 82.3 ± 5.1
15.35 ± 4.65 0
1.043 0.927
GS-MSE GS-AdaPM
1
12.04 ± 0.90 23.37 ± 1.07
13.4 ± 3.8 58.2 ± 4.1
48.82 ± 12.38 7.47 ± 1.50
6.874 1.438
GS-MSE GS-AdaPM
5
14.62 ± 0.79 23.61 ± 1.08
19.9 ± 3.5 60.1 ± 4.1
33.03 ± 6.65 7.19 ± 1.30
5.204 1.002
GS-MSE GS-AdaPM
25
24.98 ± 1.48 26.75 ± 1.96
67.0 ± 4.6 78.0 ± 4.5
3.74 ± 0.84 2.18 ± 0.50
0.989 0.985
GS-MSE GS-AdaPM
50
25.24 ± 1.56 26.85 ± 1.99
68.6 ± 4.6 78.4 ± 4.6
3.80 ± 0.82 2.08 ± 0.49
0.991 0.985
with five target images. As the adaptation set grows, both methods approach the target-trained reference and their performance gap narrows. The GS results show a larger difference between the two adaptation objectives in the low-data regime. With one and five target images, GS-MSE falls below the source-model PSNR, while GS-AdaPM achieves 23.37 and 23.61 dB. These results correspond to gains of 11.33 and 8.99 dB over GS-MSE. With 25 and 50 target images, GS-AdaPM improves PSNR by 1.77 and 1.61 dB, respectively. It also produces lower relative trajectory mismatch at every adaptation budget. For one-shot and five-shot adaptation, the empirical Jacobian estimates of GS-AdaPM are much smaller than those of GS-MSE. For 25 and 50 target images, both methods yield estimates close to one. These results show that the contractivity-regularized objective better controls the empirical local Jacobian during few-shot adaptation.
18
Figure 5: Representative ×4 super-resolution results for nadapt = 50 under the BreCaHAD-to-CelebA domain shift. The proximal-matching variants recover sharper facial contours and fewer local distortions than the corresponding MSE-adapted models. The boxes mark the enlarged regions shown below each reconstruction.
19
The visual comparisons in Figure 5 follow the quantitative results. The source-trained models produce strong smoothing and visible artifacts around facial contours. Both adaptation objectives improve the reconstructions, but the proximal-matching variants preserve sharper boundaries and more coherent local structure than the corresponding MSE baselines. Together with the deblurring results, these experiments show the same overall pattern across two forward operators. Proximal-matching adaptation provides the largest gains when target-domain data are limited, and lower relative trajectory mismatch is associated with higher reconstruction quality. 5.4 Further analysis Figure 6 examines the effect of LPN adaptation along the PnP-PGD trajectory. At the same adaptation budget, PM-adapted denoisers generally achieve higher PSNR across iterations than their MSE-adapted counterparts. The difference is largest with one or five target images and decreases as the adaptation set grows. This trajectory-level trend agrees with the final reconstruction results in Tables 2 and 3. The right-hand panels of Figure 6 compare final PSNR with the average relative trajectory mismatch. Across both inverse problems, lower mismatch generally coincides with higher reconstruction PSNR. At each adaptation budget, proximal matching typically moves the result toward the upper-left region of the plot, corresponding to lower mismatch and higher PSNR. The same pattern appears for deblurring and superresolution, supporting relative trajectory mismatch as a diagnostic of adaptation behavior.
20 15 10
Mean PSNR (dB)
1 5 10 15 20
1
5
10
30 25
30 1 5 10 15 20
20 15
15 20 PnP iteration
1
5
10
(e) SR PM convergence
zoom
zoom
28
25
27
20
26
1
10
20
30
40
15
28
25
27 26
20
1
10
25
20
30
40
15
(c) Deblurring gap vs PSNR
5
20 15 10
15 20 PnP iteration
(d) SR MSE convergence
25 50 25 5 50
30
zoom
Final PSNR (dB)
30
32
Final PSNR (dB)
25
(b) Deblurring PM convergence
zoom
Mean PSNR (dB)
32
Mean PSNR (dB)
Mean PSNR (dB)
(a) Deblurring MSE convergence 30
25 20
1
0.0
0.1
1
0.2 0.3 0.4 0.5 Mean trajectory gap
0.6
(f) SR gap vs PSNR
25 50 5 25 50 5
15
1
10
10 1
10
20 PnP iteration
Target ref.
30
40
MSE scatter
1
10
20 PnP iteration
PM scatter
30
n=1
40
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Mean trajectory gap
n=5
0.7
n=25
1
n=50
Figure 6: Trajectory-level analysis of LPN adaptation for Gaussian deblurring and ×4 super-resolution. The left and middle columns report mean PnP-PGD PSNR as a function of iteration for MSE- and PM-adapted denoisers, respectively. The right column plots final PSNR against the average relative trajectory mismatch. Proximal matching generally produces higher-PSNR trajectories and shifts the final results toward lower mismatch and higher reconstruction quality. Curves and points are averaged over 100 test images.
20
Additional qualitative comparisons across adaptation budgets are provided in Appendix C.1. For LPN-PM, the largest visual improvement occurs between one and five target images, followed by smaller refinements at larger budgets. GS-AdaPM changes less between one and five images and improves more clearly when the adaptation budget increases from five to 25 images. These observations agree with the quantitative trends in both inverse problems.
6. Conclusion We studied domain shift in PnP-PGD when the deployed denoiser differs from a proximal map associated with the target domain. We formalized this discrepancy as proximal mismatch and showed that it turns each deployed denoising update into an inexact proximal step for the target objective. Under regularity of the target proximal subproblems, we derived a nonasymptotic stationarity bound with an O(1/K) optimization term and an additive term proportional to the average squared proximal mismatch. The bound identifies persistent denoiser mismatch as a source of residual stationarity error. Guided by this result, we proposed few-shot adaptation based on proximal matching rather than MSE alone. We applied the same principle to Learned Proximal Networks and Gradient-Step denoisers, with contractivity regularization for the latter. Experiments on Gaussian deblurring and ×4 super-resolution under a BreCaHAD-to-CelebA domain shift showed that proximal matching improves reconstruction over MSEbased adaptation across both denoiser families. The gains were largest when only a few target-domain images were available and were accompanied by lower relative trajectory mismatch with the corresponding target-trained references. These results suggest that adapting a PnP denoiser should account for its role inside the reconstruction algorithm, rather than treating it only as a stand-alone denoising model. Our analysis assumes that the target prior admits a structured proximal reference, while the experiments use a target-trained reference to evaluate mismatch. Extending proximal matching to settings without such a reference, incorporating query points generated by the reconstruction trajectory, and evaluating broader domain shifts and inverse problems are important directions for future work.
21
Appendix A. Proofs A.1 Proof of Lemma 3.1 Proof Let u⋆k = D⋆ (z k ).
b k ), b k = D(z u
b k and u⋆k , By LH -smoothness of Hk on the segment between u b k − u⋆k + Hk (b uk ) ≤ Hk (u⋆k ) + ∇Hk (u⋆k ), u
LH ∥b uk − u⋆k ∥2 . 2
Since ∇Hk (u⋆k ) = 0, this gives ϵk = Hk (b uk ) − Hk (u⋆k ) ≤
LH LH 2 ∥b uk − u⋆k ∥2 = d , 2 2 k
which is (15).
A.2 Proof of Lemma 3.2 Proof By definition of the proximal objective error, ϵk+1 = Hk+1 (xk+1 ) − Hk+1 (D⋆ (z k+1 )). Since D⋆ (z k+1 ) minimizes Hk+1 , Hk+1 (D⋆ (z k+1 )) ≤ Hk+1 (xk ). Therefore, Hk+1 (xk+1 ) ≤ Hk+1 (xk ) + ϵk+1 . Expanding Hk+1 yields 1 1 R⋆ (xk+1 ) + ∥xk+1 − z k+1 ∥2 ≤ R⋆ (xk ) + ∥xk − z k+1 ∥2 + ϵk+1 . 2 2 Using z k+1 = xk − η∇f (xk ), we have xk − z k+1 = η∇f (xk ), and xk+1 − z k+1 = xk+1 − xk + η∇f (xk ). Hence 1 1 1 ∥xk − z k+1 ∥2 − ∥xk+1 − z k+1 ∥2 = −η xk+1 − xk , ∇f (xk ) − ∥xk+1 − xk ∥2 . 2 2 2
22
Thus R⋆ (xk+1 ) ≤ R⋆ (xk ) − η xk+1 − xk , ∇f (xk ) 1 − ∥xk+1 − xk ∥2 + ϵk+1 . 2
(46)
By Lf -smoothness of f , ηf (xk+1 ) ≤ ηf (xk ) + η xk+1 − xk , ∇f (xk ) +
ηLf ∥xk+1 − xk ∥2 . 2
Adding (46) and (47) gives F⋆ (xk+1 ) ≤ F⋆ (xk ) −
1 − ηLf ∥xk+1 − xk ∥2 + ϵk+1 , 2
which proves (18).
A.3 Proof of Lemma 3.3 Proof Let r k+1 := −∇Hk+1 (xk+1 ). Since
1 Hk+1 (u) = ∥u − z k+1 ∥2 + R⋆ (u), 2
we have ∇Hk+1 (xk+1 ) = xk+1 − z k+1 + ∇R⋆ (xk+1 ). Therefore, ∇R⋆ (xk+1 ) = z k+1 − xk+1 − r k+1 . This proves the residual identity in (19). For the norm bound, Assumption 3.3 gives ∥∇Hk+1 (xk+1 )∥2 ≤ 2LH Hk+1 (xk+1 ) − Hk+1 (D⋆ (z k+1 )) = 2LH ϵk+1 . Since ∥r k+1 ∥ = ∥∇Hk+1 (xk+1 )∥, the result follows.
A.4 Proof of Theorem 3.1 Proof By Lemma 3.3, ∇F⋆ (xk+1 ) = η∇f (xk+1 ) + ∇R⋆ (xk+1 ) = xk − xk+1 + η ∇f (xk+1 ) − ∇f (xk ) − r k+1 .
23
(47)
Using the Lipschitz continuity of ∇f , xk − xk+1 + η ∇f (xk+1 ) − ∇f (xk ) ≤ (1 + ηLf )∥xk+1 − xk ∥. Thus, ∥∇F⋆ (xk+1 )∥2 ≤ 2(1 + ηLf )2 ∥xk+1 − xk ∥2 + 2∥r k+1 ∥2 . By Lemma 3.3, 2∥r k+1 ∥2 ≤ 4LH ϵk+1 . By Lemma 3.2, ∥xk+1 − xk ∥2 ≤
2 F⋆ (xk ) − F⋆ (xk+1 ) + ϵk+1 . 1 − ηLf
Combining the last three displays gives ∥∇F⋆ (xk+1 )∥2 ≤
4(1 + ηLf )2 F⋆ (xk ) − F⋆ (xk+1 ) 1 − ηLf ! 4(1 + ηLf )2 + 4LH ϵk+1 . + 1 − ηLf
Since ηLf < 1, (1 + ηLf )2 ≤ 4, and hence 16 ∥∇F⋆ (xk+1 )∥2 ≤ F⋆ (xk ) − F⋆ (xk+1 ) + 1 − ηLf
16 + 4LH 1 − ηLf
! ϵk+1 .
Summing over k = 0, . . . , t − 1 yields t−1 X
t−1 16 X F⋆ (xk ) − F⋆ (xk+1 ) ∥∇F⋆ (xk+1 )∥ ≤ 1 − ηLf k=0 k=0 ! t−1 X 16 + 4LH ϵk+1 . + 1 − ηLf 2
k=0
The objective terms telescope, and Assumption 3.2 gives t−1 X
F⋆ (xk ) − F⋆ (xk+1 ) = F⋆ (x0 ) − F⋆ (xt ) ≤ F⋆ (x0 ) − F⋆inf .
k=0
After reindexing, t X k=1
16 F⋆ (x0 ) − F⋆inf + ∥∇F⋆ (xk )∥2 ≤ 1 − ηLf
Lemma 3.1 gives ϵk ≤
LH 2 d . 2 k
24
16 + 4LH 1 − ηLf
!
t X k=1
ϵk .
(48)
Substituting this into (48) and dividing by t proves (21). The bound on the minimum follows from t
min ak ≤
1≤k≤t
for nonnegative ak . Finally, if
1X ak t k=1
P∞
2 k=1 dk < ∞, then ∞ X
ϵk < ∞.
k=1
Letting t → ∞ in (48) gives
∞ X
∥∇F⋆ (xk )∥2 < ∞.
k=1
Therefore ∥∇F⋆ (xk )∥ → 0.
A.5 Proof of Corollary 3.1 b = D⋆ , then Proof If D b k ) − D⋆ (z k )∥ = 0 dk = ∥D(z for all visited query points. Lemma 3.1 gives ϵk ≤
LH 2 d = 0. 2 k
Since ϵk ≥ 0, we have ϵk = 0. Substituting dk = 0 into Theorem 3.1 gives t 1X 16 ∥∇F⋆ (xk )∥2 ≤ F⋆ (x0 ) − F⋆inf , t (1 − ηLf )t k=1
which is (22).
A.6 Proof of Proposition 4.1 Proof For a fixed y, the conditional MSE risk decomposes as h i 2 E ∥D(Y ) − X∥22 Y = y = D(y) − E[X | Y = y] 2 h i 2 + E X − E[X | Y = y] 2 Y = y . The second term is independent of D(y). Hence, D⋆MSE (y) = E[X | Y = y]. 25
The conditional PM risk equals 1 − E exp
∥D(y) − X∥22 − 2γ 2
!
Y = y .
Its minimizers therefore maximize ∥z − x∥22 exp − 2γ 2
Z z 7→
! ptar (x | y) dx.
Up to normalization, this is the Gaussian smoothing of ptar (· | y). The consistency result of Fang et al. (2024) gives D⋆PM,γ (y) −→ arg max ptar (x | y) x
as γ ↓ 0, under the stated conditions. Finally, ∥y − x∥22 ptar (x | y) ∝ exp − 2 2σadapt
! ptar (x).
Using Rtar (x) = − log ptar (x), arg max ptar (x | y) = proxσ2
adapt Rtar
x
(y).
Appendix B. Proximal Characterizations of Structured Denoisers This appendix collects the proximal characterizations used in Section 3.4 and derives the corresponding smoothness constants for the proximal subproblems. B.1 Learned Proximal Networks Recall the LPN parameterization Dθ = ∇Ψθ ,
Ψθ (x) = ψθ (x) +
α ∥x∥2 , 2
where ψθ is input-convex and α ∈ (0, 1). Proposition B.1 ((Fang et al., 2024)) Let Dθ = ∇Ψθ be an LPN as in (23). Then: (i) Dθ : Rn → Rn is a continuous bijection with continuous inverse ⋆ D−1 θ = ∇Ψθ ,
and D−1 θ is 1/α-Lipschitz; 26
(ii) there exists a real-valued, C 1 , coercive function Rθ : Rn → R such that Dθ = proxRθ ; (iii) for every x ∈ Rn ,
∇Rθ (x) = D−1 θ (x) − x.
Proximal-subproblem regularity.
Let D⋆ = Dθ⋆ ,
R⋆ = Rθ⋆ .
For the proximal subproblem 1 Hk (u) = R⋆ (u) + ∥u − z k ∥2 , 2 Proposition B.1 gives ∇Hk (u) = ∇R⋆ (u) + u − z k = D−1 θ⋆ (u) − z k . Since ⋆ D−1 θ⋆ = ∇Ψθ⋆ ,
integration yields Hk (u) = Ψ⋆θ⋆ (u) − ⟨z k , u⟩. The function Hk is convex, and its gradient is 1/α-Lipschitz. Thus Assumption 3.3 holds with LH =
1 . α
Applying Lemma 3.1 with u⋆k = Dθ⋆ (z k ),
b θ (z k ), uk = D gives ϵk ≤
1 ∥uk − u⋆k ∥2 , 2α
which is precisely (24). B.2 Gradient-Step Proximal Denoisers Recall that a GS denoiser is defined by 1 hσ (x) = ∥x∥2 − gσ (x). 2
Dσ = Id −∇gσ = ∇hσ ,
Proposition B.2 (GS proximal interpretation; (Hurault et al., 2022b)) Let X ⊂ Rn be open and convex, and let gσ : X → R be sufficiently smooth. If ∇gσ is L-Lipschitz with L < 1, then: (i) hσ is (1 − L)-strongly convex; 27
(ii) Dσ is injective; (iii) there exists an induced regularizer ϕσ such that Dσ = proxϕσ (iv) for every x ∈ Im(Dσ ),
on Im(Dσ );
∇ϕσ (x) = D−1 σ (x) − x;
(v) D−1 σ is 1/(1 − L)-Lipschitz. Proximal-subproblem regularity.
Let D⋆ = Dσ,⋆ ,
R⋆ = ϕσ,⋆ .
Because the induced regularizer is defined through the image of the target denoiser, assume that b σ (z k ) ∈ Im(Dσ,⋆ ) D for every visited query point. This ensures that ϕσ,⋆ is finite and differentiable at each produced iterate. For
1 Hk (u) = ϕσ,⋆ (u) + ∥u − z k ∥2 , 2
we have, on Im(Dσ,⋆ ), ∇Hk (u) = ∇ϕσ,⋆ (u) + u − z k = D−1 σ,⋆ (u) − z k . Since ⋆ D−1 σ,⋆ = ∇hσ,⋆ ,
it follows, up to an additive constant, that Hk (u) = h⋆σ,⋆ (u) − ⟨z k , u⟩. Because hσ,⋆ is (1 − L)-strongly convex, its conjugate has a 1/(1 − L)-Lipschitz gradient. Hence Hk is convex and 1/(1 − L)-smooth, and 1 LH = . 1−L Applying Lemma 3.1 with b σ (z k ), uk = D u⋆k = Dσ,⋆ (z k ), gives ϵk ≤
1 ∥uk − u⋆k ∥2 , 2(1 − L)
which is (26).
28
Appendix C. Additional Experimental Results C.1 Effect of the Adaptation Budget This section provides qualitative comparisons across different numbers of target-domain adaptation images. The corresponding quantitative results are reported in Tables 2 and 3. The figures show that the effect of the adaptation budget differs between the two denoiser families. LPN-PM improves most strongly between one and five target images, whereas GS-AdaPM shows a larger change between five and 25 images.
Figure 7: Representative Gaussian deblurring results for LPN-PM at different target-domain adaptation budgets. The largest visual improvement occurs between one and five adaptation images. Additional target images further reduce artifacts and refine facial contours and local details.
29
Figure 8: Representative Gaussian deblurring results for GS-AdaPM at different target-domain adaptation budgets. The reconstructions change less between one and five images and improve more clearly when 25 target images are used. Larger adaptation sets recover sharper facial contours and finer local structure.
Figure 9: Representative ×4 super-resolution results for LPN-PM at different target-domain adaptation budgets. The largest visual improvement occurs between one and five target images. Further adaptation produces smaller refinements in facial boundaries and local detail.
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Figure 10: Representative ×4 super-resolution results for GS-AdaPM at different target-domain adaptation budgets. The main visual improvement occurs when the adaptation budget increases from five to 25 images. Larger budgets produce more coherent facial boundaries and fewer local distortions.
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