Proximal Denoiser for Convergent Plug-and-Play Optimization with Nonconvex Regularization

Motivation

• Ill-posed inverse problems require regularization, yet designing deep-image regularizers that support efficient, provably convergent optimization remains open.The data term models degraded observations, while the regularizer supplies a prior to handle non-invertible operators such as those in deblurring or inpainting.Source: §1 · p. 1

Plug-and-Play methods

• PnP methods replace a regularizer’s proximal operator with a deep denoiser, achieving strong restoration results but generally lacking explicit objectives and convergence guarantees.Because generic deep denoisers need not be proximal mappings, the implicit regularizer limits interpretation and numerical monitoring.Source: §1 · p. 2

Gradient-step denoiser

• The paper studies a gradient-step denoiser parameterized by a differentiable neural network whose PnP-HQS use already converges to stationary points of an explicit functional.The earlier guarantee is specific to PnP-HQS and does not directly extend to other PnP frameworks.Source: §1 · p. 2

Contributions

• The main contribution proves that the gradient-step denoiser can be an exact proximal operator of a nonconvex smooth function, yielding convergence guarantees for PnP-PGD, PnP-ADMM, and PnP-DRS.These algorithms are guaranteed to converge to stationary points of an explicit functional.Source: §1 · p. 2

Nonexpansive denoisers

• Nonexpansive-denoiser analyses can impair denoising performance, exclude tasks through strong data-term assumptions, and target operators rather than explicit minimized functionals.The cited limitations affect applications including deblurring, super-resolution, and inpainting.Source: §2 · p. 2

Gradient step denoisers

• Gradient-step denoisers provide explicit-functional convergence for PnP-HQS, but that strategy does not extend directly to PnP-PGD or PnP-ADMM.Earlier approaches train the denoiser as a gradient step on a possibly nonconvex function.Source: §2 · p. 2

Proximal denoisers

• Classical proximal-characterization results tie proximal mappings to nonexpansiveness, motivating an extension to deep denoisers that may violate this condition.The paper pursues proximal representations based on nonconvex functionals rather than restricting denoisers to the convex case.Source: §2 · pp. 2–3
• The paper establishes convergence of PnP-PGD, PnP-ADMM, and PnP-DRS toward stationary points of explicit functions when the denoiser is learned as a proximal operator of a nonconvex regularizer.This extends proximal-based convergence analysis to deep image denoisers that are not necessarily nonexpansive.Source: §2 · p. 3

Gradient-step denoiser

• The gradient-step denoiser is written as Dσ = Id −∇gσ, with gσ generated through a differentiable neural-network parameterization.The construction uses a scalar function whose gradient defines the denoising residual.Source: §3 · p. 3

Proximal characterization

• Characterization results show that, under suitable smoothness and contractivity conditions, this gradient denoiser can equal a single-valued proximal operator of a smooth function.The resulting potential may be nonconvex, extending Moreau-type proximal representations beyond nonexpansive denoisers.Source: §3 · p. 3
• The induced proximal mapping is one-to-one and may be nonexpansive only optionally, corresponding to prox-regularity of the nonconvex potential.The paper explicitly allows Dσ to be nonexpansive or not.Source: §3 · p. 3

Implication for PnP

• Replacing a proximal step with a denoiser satisfying these conditions turns PnP-PGD and related schemes into classical proximal algorithms with a nonconvex function involved.This provides the structural basis for convergence analysis of the resulting PnP methods.Source: §3 · p. 3

4. Convergence analysis of PnP methods with proximal denoisers

• Under suitable parameter, boundedness, and KL conditions, PnP methods converge in function values, while bounded iterates converge to stationary points.The general analysis assumes a C2 denoiser potential with L-Lipschitz gradient, L < 1, and lower-bounded data and denoiser potentials.Source: §4 · p. 3

4.1. Convergence of PnP-PGD

• PnP-PGD has non-increasing objective values and stationary cluster points when f has Lipschitz gradient and λLf < 1.If the objective is KL and the iterates are bounded, the whole sequence converges with finite length to a stationary point.Source: §4.1 · p. 4

4.2.2. NON DIFFERENTIABLE DATA-FIDELITY TERM

• PnP-PGD and differentiable-data PnP-DRS retain the restriction λLf < 1, whereas the non-differentiable-data result removes this restriction.The latter advantage comes with the practically difficult assumption that Im(Dσ) is convex.Source: §4.1 · p. 4, §4.2.1 · p. 4, §4.2.2 · p. 5

4.2.1. DIFFERENTIABLE DATA-FIDELITY TERM

• For differentiable convex f, PnP-DRS and equivalently PnP-ADMM converge toward stationary points under the same main assumptions as PnP-PGD.When the generated sequence is bounded and semi-algebraic, the full sequence converges and the y and z iterates approach the same stationary point.Source: §4.2.1 · pp. 4–5, §4.2 · p. 4

4.2.2. NON DIFFERENTIABLE DATA-FIDELITY TERM

• With non-differentiable f, an inverted DRS ordering yields convergence for every λ > 0 when Im(Dσ) is convex.The result permits proper lower-semicontinuous, potentially nonconvex data fidelity, but requires convexity of the denoiser image.Source: §4.2.2 · p. 5

5.1. Proximal GS denoiser

• The proximal denoiser trades denoising performance against the Lipschitz constraint, while µ = 10^-2 retains performance comparable to FFDNet and DnCNN.Larger µ and higher noise levels worsen PSNR, but the constrained denoiser still provides high-quality denoising.Source: §5.1 · p. 6
• With sufficiently strong penalization, prox-DRUNet satisfies the L < 1 constraint across all studied CBSD68 noise levels, unlike GS-DRUNet.The constraint is evaluated through the maximum Lipschitz-related quantity S on the testing images.Source: §5.1 · p. 6

5.2.1. DEBLURRING

• Prox-PnP-DRS gives the best results among the proposed methods and remains competitive with IRCNN and DPIR while retaining convergence guarantees.Its convergence guarantee holds for any λ, enabling λ to be tuned for performance; DPIR is empirically shown to fail to converge in practice.Source: §5.2.1 · p. 7, Table 3 · p. 7

5.2.2. SUPER-RESOLUTION

• All three Prox-PnP algorithms converge on deblurring and super-resolution, while Prox-PnP-DRS matches the best super-resolution performance across scales and noise levels.For deblurring, Prox-PnP-PGD and Prox-PnP-DRS almost always reach the same local minima; for super-resolution, the methods reach DPIR-level performance.Source: §5.2.2 · pp. 7, 9, §5.2.1 · p. 7, Table 4 · p. 7

• The proximal denoiser makes PnP-PGD, PnP-ADMM, and PnP-DRS converge to stationary points of an explicit functional.PnP-PGD also handles data-fidelity terms whose proximal mappings lack closed forms.Source: §6 · p. 9

• The gradient-step denoiser can be expressed as the proximal mapping of an explicit nonconvex function, extending characterization results beyond nonexpansive denoisers.The construction relies on proximity-operator results from Gribonval and Nikolova.Source: A. Proof of Proposition 3.1 · p. 12
• When the gradient of gσ is L-Lipschitz with L < 1, the associated potential hσ is strongly convex, making the denoiser Jacobian positive definite.This positivity implies injectivity and enables defining the inverse denoiser on its image.Source: A. Proof of Proposition 3.1 · p. 12
• The constructed function φσ has a unique global proximal minimizer at Dσ(y), so Dσ = Proxφσ.The proof establishes the minimizer through the auxiliary function Ψ and the denoiser image.Source: A. Proof of Proposition 3.1 · pp. 12–13

• The objective sequence Fλ,σ(xk) is non-increasing and converges to a finite limit under the theorem’s lower-boundedness assumption.The iterates also have vanishing successive differences.Source: B. Proof of Theorem 4.1 · p. 14
• A fixed point of Tλ,σ = Proxφσ ◦(Id −λ∇f) is equivalent to a stationary point of Fλ,σ = λf + φσ.This identifies the optimization target of PnP-PGD.Source: B. Proof of Theorem 4.1 · pp. 14–15
• Every cluster point is a fixed point of the PnP-PGD operator and therefore a stationary point of Fλ,σ.The argument uses continuity and the vanishing update differences.Source: B. Proof of Theorem 4.1 · p. 15
• Full iterate convergence follows by applying KL-based convergence theory, with φσ required to be KL on Im(Dσ).This is obtained when gσ is assumed semi-algebraic.Source: B. Proof of Theorem 4.1 · p. 15

C. Proofs of PnP-DRS convergence theorems

• Under the DRS assumptions, the iterates’ cluster points are stationary for the explicit functional, and bounded KL sequences converge to a stationary point.The proof imports descent, vanishing-residual, and cluster-point results from Themelis and Patrinos (2020).Source: C. Proofs of PnP-DRS convergence… · pp. 15–16

C.1. Proof of Theorem 4.3

• For PnP-DRSdiff, applying the DRS theorem with f1 = λf and f2 = φσ requires λLf < 1.This follows from convexity of f and the theorem’s stepsize condition.Source: §C.1 · p. 16

C.2. Proof of Theorem 4.4

• The PnP-DRS proof handles φσ being smooth only on Im(Dσ) by proving semiconvexity and transferring descent arguments to that image.The image is treated as convex, while the denoiser supplies the required Lipschitz properties.Source: §C.2 · pp. 16–17
• Because a limit of yk need not lie in Im(Dσ), the proof uses cluster-point arguments to show y∗ = z∗ is stationary for Fλ,σ.This replaces the direct argument used when the limit remains in the denoiser image.Source: §C.2 · pp. 17–18

D. On the boundedness of the generates sequences

• The convergence theorems assume bounded generated sequences; coercivity of Fλ,σ is sufficient, with Lipschitz proximal or denoising maps transferring boundedness across variables.A coercive penalization can be added to enforce this property, and experiments report that it is typically inactive.Source: D. On the boundedness of the generates… · p. 18

• DRUNet light is the denoiser architecture used to parameterize Nσ.Source: E. DRUNet architecture · p. 19, Figure 3 · p. 19

F. On the performance of nonexpansive denoisers

• µ = 10^-2 is required to make the denoiser Lipschitz constant lower than 1, but this setting severely degrades denoising performance.Source: F. On the performance of nonexpansive… · p. 19
• Nonexpansiveness on the denoiser residual Id − Dσ is less harmful to denoising performance than imposing it directly on the denoiser.Source: F. On the performance of nonexpansive… · p. 19

G.1. Blur and anti-aliasing kernels

• The deblurring and super-resolution evaluations use 10 and 4 blur kernels, respectively.Source: Figure 4 · p. 20, Figure 5 · p. 20

G.2. Regularization parameters

• The same parameters are used for deblurring and super-resolution across kernels and scales, scaled only with input noise level ν.Source: §G.2 · p. 20, Table 7 · p. 20

G.3. Additional deblurring and super-resolution experiments

• Prox-PnP methods converge to visually coherent outputs, while Prox-PnP-DRS reaches or outperforms DPIR and restores sharp images.Source: §G.3 · p. 20, Figure 6 · p. 21, Figure 7 · p. 21

G.4. Robustness to initialization

• The methods are robust to small initialization perturbations, whereas stronger perturbations lead to different local minima; PnP-DRS remains consistent with larger λ.Source: §G.4 · p. 22, Table 8 · p. 22

G.5. Convergence comparison with DPIR

• After 2000 iterations, DPIR’s PSNR and residual norm do not converge in practice, unlike the proposed PnP methods.Source: §G.5 · p. 22, Figure 8 · p. 22

• With masking probability p = 0.5, PnP-DRS restores the input images with high accuracy.Source: §G.6 · p. 22, Figure 9 · p. 23
• The experiment uses σ = 10/255 and relies on Theorem 4.4 for the non-differentiable inpainting data-fidelity term.Source: §G.6 · p. 22
• The residual convergence rate O(1/k) is empirically confirmed.Source: §G.6 · p. 22