Original: G RADIENT S TEP D ENOISER FOR CONVERGENT P LUGAND -P LAY
Revised: G RADIENT S TEP D ENOISER FOR CONVERGENT P LUGAND -P LAY

Original: Anonymous authors
Revised: Anonymous authors

Original: Paper under double-blind review
Revised: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: Plug-and-Play  methods constitute a class of iterative algorithms for imaging problems where regularization is performed by an off-the-shelf denoiser.
Revised: Plug-and-Play (PnP) methods constitute a class of iterative algorithms for imaging problems where regularization is performed by an off-the-shelf denoiser.

Original: Although Plug-and-Play methods can lead to tremendous visual performance for various image problems, the few existing convergence guarantees are based on unrealistic (or suboptimal) hypotheses on the denoiser, or limited to strongly convex data terms.
Revised: Although PnP methods can lead to tremendous visual performance for various image problems, the few existing convergence guarantees are based on unrealistic (or suboptimal) hypotheses on the denoiser, or limited to strongly convex data-ﬁdelity terms.

Original: In this work we propose a new type of Plug-and-Play methods, based on half-quadratic splitting, for which the denoiser is realized as a gradient descent step on a functional parameterized by a deep neural network.
Revised: We propose a new type of PnP method, based on half-quadratic splitting, for which the denoiser is realized as a gradient descent step on a functional parameterized by a deep neural network.

Original: Exploiting convergence results for proximal gradient descent algorithms in the non-convex setting, we show that the proposed Plug-and-Play algorithm is a convergent iterative scheme that targets stationary points of an explicit global functional.
Revised: Exploiting convergence results for proximal gradient descent algorithms in the non-convex setting, we show that the proposed PnP algorithm is a convergent iterative scheme that targets stationary points of an explicit global functional.

Original: Besides, experiments show that it is possible to learn such a deep denoiser while not compromising the performance in comparison to other state-of-the-art deep denoisers used in Plug-and-Play schemes.
Revised: Besides, experiments show that it is possible to learn such a deep denoiser while not compromising the performance in comparison to other state-of-the-art deep denoisers used in PnP schemes.

Original: We apply our proximal gradient algorithm to various ill-posed inverse problems, e.g. deblurring, super-resolution and inpainting.
Revised: We apply our proximal gradient algorithm to various ill-posed inverse problems, e g. deblurring, superresolution and inpainting.

Original: For all these applications, numerical results empirically conﬁrm the convergence results.
Revised: For all these applications, numerical results empirically conﬁrm the convergence results.

Original: Experiments also show that this new algorithm reaches state-of-the-art performance, both quantitatively and qualitatively.
Revised: Experiments also show that this new algorithm reaches state-of-the-art performance, both quantitatively and qualitatively.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Image restoration (IR) problems can be formulated as inverse problems of the form
Revised: Image restoration (IR) problems can be formulated as inverse problems of the form

Original: [Equation]
Revised: [Equation]

Original: where f is a term measuring the ﬁdelity to a degraded observation y , and g is a regularization term weighted by a parameter λ ≥ 0 .
Revised: where f is a term measuring the ﬁdelity to a degraded observation y , and g is a regularization term weighted by a parameter λ ≥ 0 .

Original: Generally, the degradation of a clean image ˆ x can be modeled by a linear operation y
Revised: Generally, the degradation of a clean image ˆ x can be modeled by a linear operation y

Original: =
Revised: =

Original: A ˆ x + ξ , where A is a degradation matrix and ξ a white Gaussian noise.
Revised: A ˆ x + ξ , where A is a degradation matrix and ξ a white Gaussian noise.

Original: In this context, theclassical maximum a posteriori (MAP) derivation relates the data term to the likelihood f ( x ) = − log p ( y | x ) = 12 σ 2 ||
Revised: In this context, the maximum a posteriori (MAP) derivation relates the data-ﬁdelity term to the likelihood f ( x ) = − log p ( y | x ) = 12 σ 2 ||

Original: Ax − y
Revised: Ax − y

Original: || 2 , while the regularization term is related to the chosen prior.
Revised: || 2 , while the regularization term is related to the chosen prior.

Original: Regularization is crucial since it tackles the ill-posedness of the IR task by bringing a priori knowledge on the solution.
Revised: Regularization is crucial since it tackles the ill-posedness of the IR task by bringing a priori knowledge on the solution.

Original: A lot of research at be dedicated to designing accurate priors g .
Revised: A lot of research has been dedicated to designing accurate priors g .

Original: Among the most classical priors, one can single out total variation (Rudin et al., 1992), wavelet sparsity (Mallat, 2009) or patch-based Gaussian mixtures (Zoran & Weiss, 2011).
Revised: Among the most classical priors, one can single out total variation (Rudin et al., 1992), wavelet sparsity (Mallat, 2009) or patch-based Gaussian mixtures (Zoran & Weiss, 2011).

Original: Designing a relevant prior g is a difﬁcult task and recent approaches rather apply deep learning techniques to directly learn a prior from a database of clean images (Lunz et al., 2018; Prost et al., 2021; Gonz´alez et al., 2021).
Revised: Designing a relevant prior g is a difﬁcult task and recent approaches rather apply deep learning techniques to directly learn a prior from a database of clean images (Lunz et al., 2018; Prost et al., 2021; Gonz´alez et al., 2021).

Original: Generally, the problem (1) does not have a closed-form solution, and an optimization algorithm is required.
Revised: Generally, the problem (1) does not have a closed-form solution, and an optimization algorithm is required.

Original: First-order proximal splitting algorithms (Combettes & Pesquet, 2011) operate individually on f and g via the proximity operator
Revised: First-order proximal splitting algorithms (Combettes & Pesquet, 2011) operate individually on f and g via the proximity operator

Original: [Equation]
Revised: [Equation]

Original: Among them, half-quadratic splitting (HQS) consists in alternately applying the proximal operators of f and g .
Revised: Among them, half-quadratic splitting (HQS) (Geman & Yang, 1995) alternately applies the proximal operators of f and g .

Original: Proximal methods are particularly useful when either f or g is nonsmooth.
Revised: Proximal methods are particularly useful when either f or g is nonsmooth.

Original: Plug-and-Play (PnP) methods (Venkatakrishnan et al., 2013) build on proximal splitting algorithms by replacing the proximity operator of g with an external denoiser D , e.g. a pretrained deep network.
Revised: Plug-and-Play (PnP) methods (Venkatakrishnan et al., 2013) build on proximal splitting algorithms by replacing the proximity operator of g with a generic denoiser, e.g. a pretrained deep network.

Original: These methods achieve state-of-the-art results (Meinhardt et al., 2017; Zhang et al., 2021; Yuan et al., 2020) in various IR problems.
Revised: These methods achieve state-of-the-art results (Buzzard et al., 2018; Ahmad et al., 2020; Yuan et al., 2020; Zhang et al., 2021) in various IR problems.

Original: However, since an external denoiser cannot generally be expressed as an exact proximal mapping (Moreau, 1965), convergence results, which stem from the properties of the proximal operator, are difﬁcult to obtain.
Revised: However, since a generic denoiser cannot generally be expressed as a proximal mapping (Moreau, 1965), convergence results, which stem from the properties of the proximal operator, are difﬁcult to obtain.

Original: Moreover, the regularizer g is only made implicit via the denoising operation.
Revised: Moreover, the regularizer g is only made implicit via the denoising operation.

Original: Therefore, PnP algorithms do not seek the minimization of an explicit objective functional which strongly limits their interpretation and numerical control.
Revised: Therefore, PnP algorithms do not seek the minimization of an explicit objective functional which strongly limits their interpretation and numerical control.

Original: In order to keep tractability of a minimization problem, Romano et al.
Revised: In order to keep tractability of a minimization problem, Romano et al.

Original: (2017) proposed, with regularization by denoising (RED), an explicit prior g that exploits a given external denoiser D in the form g ( x ) = 12 (cid:104) x, x − D ( x ) (cid:105) .
Revised: (2017) proposed, with regularization by denoising (RED), an explicit prior g that exploits a given generic denoiser D in the form g ( x ) = 12 (cid:104) x, x − D ( x ) (cid:105) .

Original: With strong assumptions on the denoiser (in particular a symmetric
Revised: With strong assumptions on the denoiser (in particular a symmetric

Original: Jacobian assumption)
Revised: Jacobian assumption)

Original: , they show that it veriﬁes
Revised: , they show that it veriﬁes

Original: [Equation]
Revised: [Equation]

Original: Such a denoiser is then plugged in gradient-based minimization schemes.
Revised: Such a denoiser is then plugged in gradient-based minimization schemes.

Original: Despite having shown very good results on various image restoration tasks, as later pointed out by Reehorst & Schniter (2018) or Saremi (2019), existing deep denoisers lack Jacobian symmetry.
Revised: Despite having shown very good results on various image restoration tasks, as later pointed out by Reehorst & Schniter (2018) or Saremi (2019), existing deep denoisers lack Jacobian symmetry.

Original: Hence, RED does not minimize an explicit functional and is not guaranteed to converge.
Revised: Hence, RED does not minimize an explicit functional and is not guaranteed to converge.

Original: Contributions.
Revised: Contributions.

Original: In this work, we develop aconvergent PnP scheme with novel theoretical  guarantees and state-of-the-art image restoration performance.
Revised: In this work, we develop a PnP scheme with novel theoretical convergence guarantees and state-of-the-art IR performance.

Original: Departing froma slight modiﬁcation of the PnP-HQS framework, we plug a denoiser that inherently satisﬁes equation 3 without sacriﬁcing the denoising performance.
Revised: Departing from the PnP-HQS framework, we plug a denoiser that inherently satisﬁes equation (3) without sacriﬁcing the denoising performance.

Original: Once plugged in a PnP framework, the resulting ﬁxed-point algorithm is guaranteed to converge to a stationary point of an explicit functional.
Revised: The resulting ﬁxed-point algorithm is guaranteed to converge to a stationary point of an explicit functional.

Original: Besides, this new PnP algorithm allows for non strongly convex data terms, and thus can address ill-posed IR tasks like deblurring, super-resolution or inpainting. 
Revised: This convergence guarantee does not require strong convexity of the data-ﬁdelity term, thus encompassing ill-posed IR tasks like deblurring, super-resolution or inpainting. [Equation]

Original: R ELATED WORKS
Revised: ELATED WORKS

Original: PnP methods have been successfully applied in the literature with various splitting schemes:
Revised: PnP methods have been successfully applied in the literature with various splitting schemes:

Original: HQS (Zhang et al., 2017b; 2021), ADMM (Romano et al., 2017; Ryu et al., 2019), Proximal Gradient Descent (PGD) (Terris et al., 2020).
Revised: HQS (Zhang et al., 2017b; 2021), ADMM (Romano et al., 2017; Ryu et al., 2019), Proximal Gradient Descent (PGD) (Terris et al., 2020).

Original: First used with classical non deep denoisers such as BM3D (Chan et al., 2016) and pseudo-linear denoisers (Nair et al., 2021; Gavaskar et al., 2021), more recent PnP approaches (Meinhardt et al., 2017; Ryu et al., 2019) rely on efﬁcient off-theshelf deep denoisers such as DnCNN (Zhang et al., 2017a).
Revised: First used with classical non deep denoisers such as BM3D (Chan et al., 2016) and pseudo-linear denoisers (Nair et al., 2021; Gavaskar et al., 2021), more recent PnP approaches (Meinhardt et al., 2017; Ryu et al., 2019) rely on efﬁcient off-the-shelf deep denoisers such as DnCNN (Zhang et al., 2017a).

Original: State-of-the-art IR results are currently obtained with denoisers that are speciﬁcally designed for subsequent integration in PnP schemes, like IRCNN (Zhang et al., 2017b) or DRUNET (Zhang et al., 2021).
Revised: State-of-the-art IR results are currently obtained with denoisers that are speciﬁcally designed to be integrated in PnP schemes, like IRCNN (Zhang et al., 2017b) or DRUNET (Zhang et al., 2021).

Original: Though providing excellent restorations, such schemes are not guaranteed to converge for all kinds of denoisers or IR tasks.
Revised: Though providing excellent restorations, such schemes are not guaranteed to converge for all kinds of denoisers or IR tasks.

Original: Designing convergence proofs for PnP algorithms is an active research topic.
Revised: Designing convergence proofs for PnP algorithms is an active research topic.

Original: Sreehari et al.
Revised: Sreehari et al.

Original: ﬁrst used the proximal theorem of Moreau (Moreau, 1965) to give sufﬁcient conditions for the denoiser to be an exact proximal map, which are applied to a pseudo-linear denoiser.
Revised: ﬁrst used the proximal theorem of Moreau (Moreau, 1965) to give sufﬁcient conditions for the denoiser to be an explicit proximal map, which are applied to a pseudo-linear denoiser.

Original: The convergence with pseudo-linear denoisers have then been extensively studied (Gavaskar & Chaudhury, 2020; Nair et al., 2021; Chan, 2019).
Revised: The convergence with pseudo-linear denoisers have then been extensively studied (Gavaskar & Chaudhury, 2020; Nair et al., 2021; Chan, 2019).

Original: However, in the PnP literature, state-of-the-art results are obtained with deep denoisers.
Revised: However, in the PnP literature, state-of-the-art results are obtained with deep denoisers.

Original: Various assumptions have been proposed to ensure the convergence of the related PnP schemes.
Revised: Various assumptions have been proposed to ensure the convergence of the related PnP schemes.

Original: With a “bounded denoiser” assumption, Chan et al.
Revised: With a “bounded denoiser” assumption, Chan et al.

Original: ; Gavaskar & Chaudhury (2019) ﬁrst showed convergence of PnP-ADMM with stepsizes decreasing to 0.
Revised: ; Gavaskar & Chaudhury (2019) ﬁrst showed convergence of PnP-ADMM with stepsizes decreasing to 0.

Original: Cohen et al.
Revised: Cohen et al.

Original: (2021) considers the wider class of demi-contractive mappings, but this condition is hard to verify in practice.
Revised: (2021) considers the wider class of demi-contractive mappings, but this condition is hard to verify in practice.

Original: Many works focus on the Lipschitz properties of the PnP operator, in order to rely on Banach ﬁxed-point theorem.
Revised: Many works focus on the Lipschitz properties of the PnP operator, in order to rely on Banach ﬁxed-point theorem.

Original: Depending on the splitting algorithm in use, convergence can then be obtained by assuming the denoiser averaged (Sun et al., 2019), ﬁrmly nonexpansive (Sun et al., 2021; Terris et al., 2020) or simply a nonexpansive (Reehorst & Schniter, 2018;
Revised: Depending on the splitting algorithm in use, convergence can then be obtained by assuming the denoiser averaged (Sun et al., 2019), ﬁrmly nonexpansive (Sun et al., 2021; Terris et al., 2020) or simply a nonexpansive (Reehorst & Schniter, 2018;

Original: Liu et al., 2021).
Revised: Liu et al., 2021).

Original: These settings are unrealistic as deep denoisers do not generally satisfy such properties.
Revised: These settings are unrealistic as deep denoisers do not generally satisfy such properties.

Original: Methods like Ryu et al.
Revised: Methods like Ryu et al.

Original: (2019) or Terris et al.
Revised: (2019) or Terris et al.

Original: (2020) thus propose different ways to train deep denoisers with constrained Lipschitz constants in order to ﬁt the technical properties required for convergence.
Revised: (2020) thus propose different ways to train deep denoisers with constrained Lipschitz constants in order to ﬁt the technical properties required for convergence.

Original: But imposing hard Lipschitz constraints on the network alters its denoising performance.
Revised: But imposing hard Lipschitz constraints on the network alters its denoising performance (Bohra et al., 2021; Hertrich et al., 2021).

Original: Yet, Ryu et al.
Revised: Yet, Ryu et al.

Original: (2019) manages to get a convergent PnP scheme without assuming the nonexpansiveness of D .
Revised: (2019) manages to get a convergent PnP scheme without assuming the nonexpansiveness of D .

Original: This comes at the cost of imposing strong convexity on the data term f , which excludes many IR tasks like deblurring, super-resolution and inpainting.
Revised: This comes at the cost of imposing strong convexity on the data-ﬁdelity term f , which excludes many IR tasks like deblurring, super-resolution and inpainting.

Original: As a matter of fact, given the ill-posedness of IR problems, looking for a unique solution via contractive operators is a restrictive assumption.
Revised: As a matter of fact, given the ill-posedness of IR problems, looking for a unique solution via contractive operators is a restrictive assumption.

Original: In this work, we do not impose contractiveness, but still obtain convergence results with realistic hypotheses.
Revised: In this work, we do not impose contractiveness, but still obtain convergence results with realistic hypotheses.

Original: One can relate the  deep denoiser to the “true” natural image prior  via Tweedie’s Identity (Efron, 2011).
Revised: One can relate the ideal deep denoiser to the “true” natural image prior p via Tweedie’s Identity.

Original: The latter shows that the Minimum Mean Square Error (MMSE) denoiser D ∗ σ (at noise level σ ) veriﬁes D σ ( x ) = x− σ 2 ∇ x log p σ ( x ) with p σ ( x ) =
Revised: In (Efron, 2011), it is indeed shown that the Minimum Mean Square Error (MMSE) denoiser D ∗ σ (at noise level σ ) veriﬁes D σ ( x ) = x

Original:  p ∗ N (0 , σ 2 Id) the Gaussian-smoothed true prior.
Revised: + σ 2 ∇ x log p σ ( x ) where p σ is the convolution of p with the density of N (0 , σ 2 Id) .

Original: In a recent line of research (Bigdeli et al., 2017; Xu et al., 2020; Laumont et al., 2021; Kadkhodaie & Simoncelli, 2020), this relation is used to plug a denoiser in gradient-based dynamics.
Revised: In a recent line of research (Bigdeli et al., 2017; Xu et al., 2020; Laumont et al., 2021; Kadkhodaie & Simoncelli, 2020), this relation is used to plug a denoiser in gradientbased dynamics.

Original: In practice, the MMSE denoiser is however intractable and Tweedie’s Identity does not hold for deep approximations of the MMSE.
Revised: In practice, the MMSE denoiser cannot be computed explicitly and Tweedie’s Identity does not hold for deep approximations of the MMSE.

Original: Finally, let us mention that the form of prior considered here was already suggested in Bigdeli & Zwicker (2017) and Romano et al.
Revised: Finally, let us mention that the form of prior considered here was already suggested in Bigdeli & Zwicker (2017) and Romano et al.

Original: (2017, Section 5.2) but not exploited for convergence analysis.
Revised: (2017, Section 5.2) but not exploited for convergence analysis.

Original:  T HE G RADIENT S TEP P LUG - AND -P LAY
Revised:  T HE G RADIENT S TEP P LUG - AND -P LAY

Original: The proposed method is based on the PnP version of half-quadratic-splitting (PnP-HQS) that amounts to replacing the proximity operator of the prior g with an off-the-shelf denoiser D σ .
Revised: The proposed method is based on the PnP version of half-quadratic-splitting (PnP-HQS) that amounts to replacing the proximity operator of the prior g with an off-the-shelf denoiser D σ .

Original: In order to deﬁne a convergent PnP scheme, we ﬁrst set up in Section 3.1 a Gradient Step (GS) denoiser.
Revised: In order to deﬁne a convergent PnP scheme, we ﬁrst set up in Section 3.1 a Gradient Step (GS) denoiser.

Original: We then introduce the Gradient Step PnP algorithm in Section 3.2.
Revised: We then introduce the Gradient Step PnP algorithm in Section 3.2.

Original: 3. G RADIENT S TEP DENOISER
Revised: 3. G RADIENT S TEP DENOISER

Original: We propose to plug a denoising operator D σ that takes the form of a gradient descent step
Revised: We propose to plug a denoising operator D σ that takes the form of a gradient descent step

Original: [Equation]
Revised: [Equation]

Original: with g σ : R n → R .
Revised: with g σ : R n → R .

Original: Contrary to Romano et al.
Revised: Contrary to Romano et al.

Original: (2017), our denoiser exactly represents a conservative vector ﬁeld.
Revised: (2017), our denoiser exactly represents a conservative vector ﬁeld.

Original: The choice of the parameterization of g σ is fundamental for the denoising performance.
Revised: The choice of the parameterization of g σ is fundamental for the denoising performance.

Original: As already noticed in Salimans & Ho (2021), we experimentally found that directly modeling g σ as a neural network ( e.g. a standard network used for classiﬁcation) leads to poor denoising performance.
Revised: As already noticed in Salimans & Ho (2021), we experimentally found that directly modeling g σ as a neural network ( e.g. a standard network used for classiﬁcation) leads to poor denoising performance.

Original: In order to keep the strength of state-of-the-art unconstrained denoisers, we rather use
Revised: In order to keep the strength of state-of-the-art unconstrained denoisers, we rather use

Original: [Equation]
Revised: [Equation]

Original: where N σ : R n → R n is parameterized by a neural network.
Revised: where N σ : R n → R n is parameterized by a neural network.

Original: The resulting denoising operator is
Revised: The resulting denoising operator is

Original: [Equation]
Revised: [Equation]

Original: ) where J N σ ( x ) refers to the Jacobian of N σ at point x .
Revised: where J N σ ( x ) refers to the Jacobian of N σ at point x .

Original: Let us highlight that, thanks to this deﬁnition of D σ , we can parameterize N σ with any differentiable neural network architecture R n → R n that has proven efﬁcient for image denoising.
Revised: Let us highlight that, thanks to this deﬁnition of D σ , we can parameterize N σ with any differentiable neural network architecture R n → R n that has proven efﬁcient for image denoising.

Original: Although the representation power of the denoiser is limited by the particular form (6), we show (see Section 5.1) that such parameterization still yields state-of-the-art denoising results.
Revised: Although the representation power of the denoiser is limited by the particular form (6), we show (see Section 5.1) that such parameterization still yields state-of-the-art denoising results.

Original: We train the denoiser D σ by minimizing the MSE loss function
Revised: We train the denoiser D σ by minimizing the MSE loss function

Original: [Equation]
Revised: [Equation]

Original: Using equation 4, it can be written in terms of g σ :
Revised: Using equation (4), it can be written in terms of g σ :

Original: [Equation]
Revised: [Equation]

Original: [Equation]

Original: Remark 1.
Revised: Remark 1.

Original: By deﬁnition, the optimal solution g ∗ σ ∈ arg min L is related to the MMSE denoiser D ∗ σ , that is, the best non-linear predictor of x given x + ξ σ .
Revised: By deﬁnition, the optimal solution g ∗ σ ∈ arg min L is related to the MMSE denoiser D ∗ σ , that is, the best non-linear predictor of x given x + ξ σ .

Original: Therefore, it satisﬁes Tweedie’s formula and ∇ g ∗ σ =  ∇ log p σ (Efron, 2011) i-e g ∗ σ =  log p σ + C , for some C ∈ R .
Revised: Therefore, it satisﬁes Tweedie’s formula and ∇ g ∗ σ = − σ 2 ∇ log p σ (Efron, 2011) i.e. g ∗ σ = − σ 2 log p σ + C , for some C ∈ R .

Original: Hence approximating the MMSE denoiser with a denoiser parameterized as (4) is related to approximating the logarithm of the smoothed image prior p σ with  g σ .
Revised: Hence approximating the MMSE denoiser with a denoiser parameterized as (4) is related to approximating the logarithm of the smoothed image prior p σ with − 1 σ 2 g σ .

Original: This relation was used for image generation in the context of “Denoising Score Matching” by Saremi & Hyvarinen (2019); Bigdeli et al.
Revised: This relation was used for image generation in the context of “Denoising Score Matching” by Saremi & Hyvarinen (2019); Bigdeli et al.

Original: 3. A P LUG - AND -P LAY METHOD FOR EXPLICIT MINIMIZATION
Revised: 3. A P LUG - AND -P LAY METHOD FOR EXPLICIT MINIMIZATION

Original: The standard PnP-HQS operator is T PnP-HQS =
Revised: The standard PnP-HQS operator is T PnP-HQS =

Original: D σ ◦ Prox τf , i.e. (Id −∇ g σ ) ◦ Prox τf when using the
Revised: D σ ◦ Prox τf , i.e. (Id −∇ g σ ) ◦ Prox τf when using the

Original: GS denoiser as D σ .
Revised: GS denoiser as D σ .

Original: For convergence analysis, it proves fruitful to ﬁt the proximal gradient descent algorithm by switching the proximal and gradient steps.
Revised: For convergence analysis, it proves fruitful to ﬁt the proximal gradient descent algorithm by switching the proximal and gradient steps.

Original: We also propose to relax the denoising step with a parameter λ ≥ 0 .
Revised: We also propose to relax the denoising step with a parameter λ ≥ 0 .

Original: Our PnP algorithm with GS denoiser (GS-PnP) then writes {+x k +
+}
Revised: Our PnP algorithm with GS denoiser (GS-PnP) then writes {+x k +
+}

Original: [Equation]
Revised: =

Original: T τ,λ GS-PnP 
Revised: T τ,λ GS-PnP ( x k ) with T τ,λ GS-PnP = Prox τf ◦

Original: [Equation]
Revised: [Equation]

Original: Under suitable conditions on f and g σ (see Lemma 1 in Appendix A), ﬁxed points of the operator T τ,λ GS-PnP correspond to critical points of
Revised: Under suitable conditions on f and g σ (see Lemma 1 in Appendix A), ﬁxed points of the operator T τ,λ GS-PnP correspond to critical points of

Original: [Equation]
Revised: [Equation]

Original: Therefore, using the GS denoiser from equation 4 comes down to including an explicit regularization and thus leads to a tractable global optimization problem solved by the PnP algorithm.
Revised: Therefore, using the GS denoiser from equation (4) is equivalent to include an explicit regularization and thus leads to a tractable global optimization problem solved by the PnP algorithm.

Original: Our complete
Revised: Our complete

Original: PnP scheme is presented in Algorithm 1.
Revised: PnP scheme is presented in Algorithm 1.

Original: It includes a backtracking procedure that will be detailed in
Revised: It includes a backtracking procedure that will be detailed in

Original: Section 4.2.
Revised: Section 4.2.

Original: Also, after convergence, we found it useful to apply an extra gradient step Id − λτ ∇ g σ in order to discard the residual noise brought by the last proximal step Prox τf .
Revised: Also, after convergence, we found it useful to apply an extra gradient step Id − λτ ∇ g σ in order to discard the residual noise brought by the last proximal step Prox τf .

Original:  C ONVERGENCE ANALYSIS
Revised:  C ONVERGENCE ANALYSIS

Original: Param.
Revised: Param.

Original: : init.
Revised: : init.

Original: z 0 , λ > 0 , σ ≥ 0 , (cid:15) > 0 , τ 0 > 0 ,
Revised: z 0 , λ > 0 , σ ≥ 0 , (cid:15) > 0 , τ 0 > 0 ,

Original: K ∈ N ∗ , η ∈ (0 , 1) , γ ∈ (0 , 1 / 2) .
Revised: K ∈ N ∗ , η ∈ (0 , 1) , γ ∈ (0 , 1 / 2) .

Original: Input : degraded image y .
Revised: Input : degraded image y .

Original: Output: restored image ˆ x . k = 0 ; x 0 =
Revised: Output: restored image ˆ x . k = 0 ; x 0 =

Original: Prox τf ( z 0 ) ; τ = τ 0 /η ; dif > (cid:15) ; while k < K and dif > (cid:15) do z k = λτD σ ( x k )
Revised: Prox τf ( z 0 ) ; τ = τ 0 /η ; dif > (cid:15) ; while k < K and dif > (cid:15) do z k = λτD σ ( x k )

Original: + (1 − λτ ) x k ; x k +

Revised: + (1 − λτ ) x k ; x k +

Original: =
Revised: =

Original: Prox τf ( z k ) ; if F ( x k ) − F ( x k +1 ) <
Revised: Prox τf ( z k ) ; if F ( x k ) − F ( x k +1 ) <

Original: γτ || x
Revised: γτ || x

Original: k − x k +

Revised: k − x k +

Original: || 

Revised: ||

Original: ; then τ = ητ ; else dif = F ( x k ) − F ( x k +1 ) F ( x 0 ) ;
Revised: ; then τ = ητ ; else dif = F ( x k ) − F ( x k +1 ) F ( x 0 ) ;

Original: k
Revised: k

Original: = k + 1 ; end ˆ x = λτD σ ( x K )
Revised: = k + 1 ; end ˆ x = λτD σ ( x K )

Original: + (1 − λτ ) x K ;
Revised: + (1 − λτ ) x K ;

Original: In this section, we introduce conditions on f and D σ that will ensure the convergence of the
Revised: In this section, we introduce conditions on f and D σ that will ensure the convergence of the

Original: PnP algorithm (9) towards a solution of (10).
Revised: PnP iterations (9) towards a solution of (10).

Original: For that purpose, we make use of the literature of convergence analysis (Attouch et al., 2013;
Revised: For that purpose, we make use of the literature of convergence analysis (Attouch et al., 2013;

Original: Beck, 2017; Beck & Teboulle, 2009; Gu et al., 2018) of the proximal gradient descent algorithm in the non-convex setting.
Revised: Beck, 2017; Beck & Teboulle, 2009; Gu et al., 2018) of the proximal gradient descent algorithm in the non-convex setting.

Original: 4. C ONVERGENCE RESULTS
Revised: 4. C ONVERGENCE RESULTS

Original: A common setting for image restoration is the convex smooth L 2 data ﬁdelity f ( x ) =
Revised: A common setting for image restoration is the convex smooth L 2 data-ﬁdelity f ( x ) =

Original: ||
Revised: ||

Original: Ax − y ||
Revised: Ax − y ||

Original: In order to cover the noiseless case or to deal with a broader range of degradation models, like Laplace or Poisson noise, we only assume f to be proper, lower semicontinous and convex.
Revised: In order to cover the noiseless case or to deal with a broader range of degradation models, like Laplace or Poisson noise, we only assume f to be proper, lower semicontinous and convex.

Original: On the other hand, the regularizer g σ is assumed to be differentiable with Lipschitz gradient, but not necessarily convex.
Revised: On the other hand, the regularizer g σ is assumed to be differentiable with Lipschitz gradient, but not necessarily convex.

Original: Without further assumptions on f and g σ , the following theorem establishes the convergence of the objective function values and convergence of the residual.
Revised: Without further assumptions on f and g σ , the following theorem establishes the convergence of the objective function values and convergence of the residual.

Original: Theorem 1 (Proof in Appendix A) .
Revised: Theorem 1 (Proof in Appendix A) .

Original: Let f : R n → R ∪ { + ∞} and g σ : R n → R be proper lower semicontinous functions with f convex and g σ differentiable with L -Lipschitz gradient.
Revised: Let f : R n → R ∪ { + ∞} and g σ : R n → R be proper lower semicontinous functions with f convex and g σ differentiable with L -Lipschitz gradient.

Original: Let λ > 0 ,
Revised: Let λ > 0 ,

Original: F = f + λg σ and assume that F is bounded from below.
Revised: F = f + λg σ and assume that F is bounded from below.

Original: Then, for τ < 1 λL , the iterates x k given by Algorithm (9) verify
Revised: Then, for τ < 1 λL , the iterates x k given by the iterative scheme (9) verify

Original: (i) ( F ( x k )) is non-increasing and converging.
Revised: (i) ( F ( x k )) is non-increasing and converges.

Original: (ii)
Revised: (ii)

Original: The residual || x k +1 − x k || converges to 0 .
Revised: The residual || x k +1 − x k || converges to 0 .

Original: (iii) All cluster points of the sequence ( x k ) are stationary points of (10).
Revised: (iii) All cluster points of the sequence ( x k ) are stationary points of (10).

Original: Remark 2.
Revised: Remark 2.

Original: In the non-convex setting, the quantity γ k = min 0 ≤ i ≤ k
Revised: In the non-convex setting, the quantity γ k = min 0 ≤ i ≤ k

Original: || x
Revised: || x

Original: i +1 − x i || 2 is commonly used to analyze the convergence rate of the algorithm (Beck & Teboulle, 2009; Ochs et al., 2014).
Revised: i +1 − x i || 2 is commonly used to analyze the convergence rate of the algorithm (Beck & Teboulle, 2009; Ochs et al., 2014).

Original: Following the proof of Theorem 1 in Appendix A, we can obtain γ k ≤ 1 k F ( x 0 ) − lim F ( x k ) 12 τ − Lthat is to say a O ( 1 k ) convergence rate for the squared L 2 residual.
Revised: Following the proof of Theorem 1 in Appendix A, we can obtain γ k ≤ 1 k F ( x 0 ) − lim F ( x k ) 12 τ − Lthat is to say a O ( 1 k ) convergence rate for the squared L 2 residual.

Original: Remark 3.
Revised: Remark 3.

Original: Even if most data terms f encountered in image restoration are convex, Theorem 1 can be extended to nonconvex f .
Revised: Even if most data-ﬁdelity terms f encountered in image restoration are convex, Theorem 1 can be extended to nonconvex f .

Original: The proof of (iii) requires technical adaptations that can be found in (Li & Lin, 2015, Theorem 1) (in particular the 1-Lipschitz property of Prox τf does not hold anymore).
Revised: The proof of (iii) requires technical adaptations that can be found in (Li & Lin, 2015, Theorem 1) (in particular the 1-Lipschitz property of Prox τf does not hold anymore).

Original: We can further obtain convergence of the iterates by assuming that the generated sequence ( x k ) is bounded and that f and g σ verify the Kurdyka-Lojasiewicz (KL) property.
Revised: We can further obtain convergence of the iterates by assuming that the generated sequence ( x k ) is bounded and that f and g σ verify the Kurdyka-Lojasiewicz (KL) property.

Original: The boundedness of ( x k ) is discussed in Appendix B.
Revised: The boundedness of ( x k ) is discussed in Appendix B.

Original: The KL property (deﬁned in Appendix C) has been widely used to study the convergence of optimization algorithms in the non-convex setting (Attouch et al., 2010; 2013;
Revised: The KL property (deﬁned in Appendix C) has been widely used to study the convergence of optimization algorithms in the non-convex setting (Attouch et al., 2010; 2013;

Original: Ochs et al., 2014).
Revised: Ochs et al., 2014).

Original: Very large classes of functions, in particular all the semi-algebraic functions, satisfy this property.
Revised: Very large classes of functions, in particular all the semi-algebraic functions, satisfy this property.

Original: In practice, in the extent of our analysis, the KL property is always satisﬁed.
Revised: In practice, in the extent of our analysis, the KL property is always satisﬁed.

Original: Theorem 2 (Proof in Attouch et al.
Revised: Theorem 2 (Proof in Attouch et al.

Original: Theorem 5.1) .
Revised: Theorem 5.1) .

Original: Let f : R n → R ∪ { + ∞} and g σ : R n → R be proper lower semicontinous functions with f convex and g σ differentiable with
Revised: Let f : R n → R ∪ { + ∞} and g σ : R n → R be proper lower semicontinous functions with f convex and g σ differentiable with

Original: L -Lipschitz gradient.
Revised: L -Lipschitz gradient.

Original: Let λ > 0 , F = f + λg σ and assume that F is bounded from below.
Revised: Let λ > 0 , F = f + λg σ and assume that F is bounded from below.

Original: Assume that f and g σ verify the KL property.
Revised: Assume that F verify the KL property.

Original: Suppose that τ < 1 λL .
Revised: Suppose that τ < 1 λL .

Original: If the sequence ( x k ) given by Algorithm (9) is bounded, then it converges, with ﬁnite length, to a critical point x ∗ of F .
Revised: If the sequence ( x k ) given by the iterative scheme (9) is bounded, then it converges, with ﬁnite length, to a critical point x ∗ of F .

Original: Remark 4.
Revised: Remark 4.

Original: As explained in Attouch et al.
Revised: As explained in Attouch et al.

Original: (2013, Remark 5.2), the continuity of f is not required since we use the “exact forward-backward splitting algorithm”.
Revised: (2013, Remark 5.2), the continuity of f is not required since we use the “exact forward-backward splitting algorithm”.

Original: We can therefore deal with noncontinuous data terms, as it is the case with the inpainting applicationpresented in Appendix G.3.
Revised: We can thus deal with non-continuous data-ﬁdelity terms, as it is the case with the inpainting application in Appendix G.4.

Original: As detailed before, the assumptions in Theorems 1 and 2 on f and g σ are reasonable from a practical perspective.
Revised: As detailed before, the assumptions in Theorems 1 and 2 on f and g σ are reasonable from a practical perspective.

Original: Indeed, using a network N σ with differentiable activation functions, the function g σ introduced in Section 3 is differentiable with Lipschitz gradient (details and proof are given in Appendix D).
Revised: Indeed, using a network N σ with differentiable activation functions, the function g σ introduced in Section 3 is differentiable with Lipschitz gradient (details and proof are given in Appendix D).

Original: Therefore, for a large variety of image restoration tasks, our PnP algorithm (9) is guaranteed to converge to a stationary point of the targeted objective function F .
Revised: Therefore, for a large variety of image restoration tasks, our PnP scheme (9) is guaranteed to converge to a stationary point of the targeted objective function F .

Original: 4. B ACKTRACKING TO HANDLE THE L IPSCHITZ CONSTANT OF ∇ g σ
Revised: 4. B ACKTRACKING TO HANDLE THE L IPSCHITZ CONSTANT OF ∇ g σ

Original: The convergence of Algorithm 1 actually requires to control the Lipschitz constant of ∇ g σ only on a small subset of images related to { x k } .
Revised: The convergence of Algorithm 1 actually requires to control the Lipschitz constant of ∇ g σ only on a small subset of images related to { x k } .

Original: Therefore, estimating L for all images and setting the maximum stepsize τ accordingly will lead to sub-optimal convergence speed.
Revised: Therefore, estimating L for all images and setting the maximum stepsize τ accordingly will lead to sub-optimal convergence speed.

Original: In order to avoid small stepsizes, we use the backtracking strategy (Beck, 2017, Chapter 10).
Revised: In order to avoid small stepsizes, we use the backtracking strategy (Beck, 2017, Chapter 10).

Original: The convergence study in the proof of Theorem 1 is based on the sufﬁcient decrease property of F established equation (22).
Revised: The convergence study in the proof of Theorem 1 is based on the sufﬁcient decrease property of F established equation (22).

Original: Without knowing the exact Lipschitz constant L , backtracking aims at ﬁnding the maximal stepsize τ that yields the sufﬁcient decrease property.
Revised: Without knowing the exact Lipschitz constant L , backtracking aims at ﬁnding the maximal stepsize τ that yields the sufﬁcient decrease property.

Original: Given γ ∈ (0 , 1 / 2) , η ∈ [0 , 1) and an initial stepsize τ 0 > 0 , the backtracking procedure consists in applying at each iteration k
Revised: Given γ ∈ (0 , 1 / 2) , η ∈ [0 , 1) and an initial stepsize τ 0 > 0 , the backtracking procedure consists in applying at each iteration k

Original: the following update rule on τ :
Revised: the following update rule on τ :

Original: [Equation]
Revised: [Equation]

Original: Proposition 1 (Proof in Appendix E) .
Revised: Proposition 1 (Proof in Appendix E) .

Original: Under the assumptions of Theorem 1, at each iteration of the algorithm, the backtracking procedure (11) is ﬁnite ( i.e. a stepsize satifying
Revised: Under the assumptions of Theorem 1, at each iteration of the algorithm, the backtracking procedure (11) is ﬁnite ( i.e. a stepsize satifying

Original: F ( x k ) − F ( T τ,λ GS-PnP ( x k )) ≥
Revised: F ( x k ) − F ( T τ,λ GS-PnP ( x k )) ≥

Original: γτ || T τ,λ GS-PnP ( x k ) − x k || 2 is found in a ﬁnite number if iterations), and with backtracking, the convergence results of Theorem 1 and Theorem 2 still hold.
Revised: γτ || T τ,λ GS-PnP ( x k ) − x k || 2 is found in a ﬁnite number if iterations), and with backtracking, the convergence results of Theorem 1 and Theorem 2 still hold.

Original: Remark 5.
Revised: Remark 5.

Original: In practice, as explained in Section 5, we choose to initialize the stepsize as λτ 0 =and backtracking hardly ever activates.
Revised: In practice, as explained in Section 5, we choose to initialize the stepsize as λτ 0 =and backtracking hardly ever activates.

Original: In this particular case, our results prove convergence of the standard PnP-HQS scheme x k +

Revised: In this particular case, our results prove convergence of the standard PnP-HQS scheme x k +

Original: =
Revised: =

Original: Prox τf ◦ D σ ( x k ) applied with our speciﬁc denoiser.
Revised: Prox τf ◦ D σ ( x k ) applied with our speciﬁc denoiser.

Original:  E XPERIMENTS 5. G RADIENT - DESCENT - BASED DENOISER
Revised:  E XPERIMENTS 5. G RADIENT - DESCENT - BASED DENOISER

Original: Denoising Network Architecture We choose to parameterize N σ with the architecture
Revised: Denoising Network Architecture We choose to parameterize N σ with the architecture

Original: DRUNet (Zhang et al., 2021)), a U-Net in which residual blocks are integrated.
Revised: DRUNet (Zhang et al., 2021)) (represented in Appendix H), a U-Net in which residual blocks are integrated.

Original: One ﬁrst beneﬁt of DRUNet is that it is built to take the noise level σ as input, which is consistent with our formulation.
Revised: One ﬁrst beneﬁt of DRUNet is that it is built to take the noise level σ as input, which is consistent with our formulation.

Original: Also, the U-Net models have previously offered good results in the context of prior approximation via Denoising Score Matching
Revised: Also, the U-Net models have previously offered good results in the context of prior approximation via Denoising Score Matching

Original: (Ho et al., 2020).
Revised: (Ho et al., 2020).

Original: Furthermore, Zhang et al.
Revised: Furthermore, Zhang et al.

Original: (2021) showed that DRUNet yields state-of-the-art results for denoising but also forsubsequent PnP image restoration.
Revised: (2021) showed that DRUNet yields state-of-the-art results for denoising but also for PnP image restoration.

Original: In order to ensure differentiability w.r.t the input, we change RELU activations to ELU.
Revised: In order to ensure differentiability w.r.t the input, we change RELU activations to ELU.

Original: We also limit the number of residual blocks to 2 at each scale to lower the computational burden.
Revised: We also limit the number of residual blocks to 2 at each scale to lower the computational burden.

Original: Training details We use the color image training dataset proposed in Zhang et al.
Revised: Training details We use the color image training dataset proposed in Zhang et al.

Original: (2021) i.e. a combination of the Berkeley segmentation dataset (CBSD) (Martin et al., 2001), Waterloo Exploration Database (Ma et al., 2017), DIV2K dataset (Agustsson & Timofte, 2017) and Flick2K dataset (Lim et al., 2017).
Revised: (2021) i.e. a combination of the Berkeley segmentation dataset (CBSD) (Martin et al., 2001), Waterloo Exploration Database (Ma et al., 2017), DIV2K dataset (Agustsson & Timofte, 2017) and Flick2K dataset (Lim et al., 2017).

Original: During training, the input images are corrupted with a white Gaussian noise ξ σ with standard deviation σ randomly chosen in [0 , 50 / 255] .
Revised: During training, the input images are corrupted with a white Gaussian noise ξ σ with standard deviation σ randomly chosen in [0 , 50 / 255] .

Original: With our parameterization (6) of D σ , the network N σ is trained to minimize the L 2 loss (8).
Revised: With our parameterization (6) of D σ , the network N σ is trained to minimize the L 2 loss (7).

Original: While the original DRUNet is trained with L 1 loss, we stick to the L 2 lossso as to keep the interpretability of g σ as an approximation of the log prior (see Remark 1).
Revised: While the original DRUNet is trained with L 1 loss, we stick to the L 2 loss to keep the interpretability of g σ as an approximation of the log prior (see Remark 1).

Original: For each batch, the gradient ∇ g σ is calculated with PyTorchautomatic differentiation tools.
Revised: For each batch, the gradient ∇ g σ is computed with PyTorch differentiation tools.

Original: We train the model on 128 × 128 patches randomly sampled from the training images, with batch size 16 , during 1500 epochs.
Revised: We train the model on 128 × 128 patches randomly sampled from the training images, with batch size 16 , during 1500 epochs.

Original: We use the ADAM optimizer with learning rate 10 − 4 , divided by 2 every 300 epochs.
Revised: We use the ADAM optimizer with learning rate 10 − 4 , divided by 2 every 300 epochs.

Original: It takes around one week to train the model on a single Tesla P100 GPU.
Revised: It takes around one week to train the model on a single Tesla P100 GPU.

Original: Denoising results We evaluate the PSNR performance of the proposed GS denoiser (GS-DRUNet) on 256 × 256 color images center-cropped from the original CBSD68 dataset images (Martin et al., 2001).
Revised: Denoising results We evaluate the PSNR performance of the proposed GS denoiser (GS-DRUNet) on 256 × 256 color images center-cropped from the original CBSD68 dataset images (Martin et al., 2001).

Original: In Table 1, we compare, for various noise levels σ , our model with the simpliﬁed DRUNet (called “DRUNet light ”) that has the same architecture as our GS-DRUNet (2 residual blocks) and that is trained (with L 2 loss) without the conservative ﬁeld constraint.
Revised: In Table 1, we compare, for various noise levels σ , our model with the simpliﬁed DRUNet (called “DRUNet light ”) that has the same architecture as our GS-DRUNet (2 residual blocks) and that is trained (with L 2 loss) without the conservative ﬁeld constraint.

Original: We also provide comparisons with the original DRUNet (Zhang et al., 2021)
Revised: We also provide comparisons with the original DRUNet (Zhang et al., 2021)

Original: (with 4 residual blocks at each scale and trained with
Revised: (with 4 residual blocks at each scale and trained with

Original: L 1 loss) and two state-of-the-art denoisers encountered in the PnP literature: FFDNet (Zhang et al., 2018) and
Revised: L 1 loss) and two state-of-the-art denoisers encountered in the PnP literature: FFDNet (Zhang et al., 2018) and

Original: DnCNN (Zhang et al., 2017a).
Revised: DnCNN (Zhang et al., 2017a).

Original: For each network, we indicate in Table 1 the average runtime while processing a 256 × 256 color image on one Tesla P100 GPU.
Revised: For each network, we indicate in Table 1 the average runtime while processing a 256 × 256 color image on one Tesla P100 GPU.

Original: [Table]
Revised: [Table]

Original: for various noise levelsσ .
Revised: for various noise levels

Original: While keeping small runtime, GS-DRUNet slightly outperforms its unconstrained counterpart DRUNet light and outdistances the deep denoisers FFDNet and DnCNN.
Revised: While keeping small runtime GS-DRUNet slightly outperforms its constrained light and outdistances the deep denoisers FFDNet and DnCNN.

Original: Our GS-DRUNet denoiser, despite being constrained to be an exact conservative ﬁeld, reaches the performance of (and even slightly outperforms) its unconstrained counterpart
Revised: Our GS-DRUNet denoiser, despite being constrained to be an exact conservative ﬁeld, reaches the performance of (and even slightly outperforms) its unconstrained counterpart

Original: DRUNet light .
Revised: DRUNet light .

Original: Second, departing from the latter, we are able to reduce the processing time by a large margin ( ÷ 7 ) while keeping close PSNR to the original DRUNet (around -0.05dB) and maintaining a signiﬁcant PSNR gap (around +0.5dB) with other deep denoisers like DnCNN and FFDNet.
Revised: Second, departing from the latter, we are able to reduce the processing time by a large margin ( ÷ 7 ) while keeping close PSNR to the original DRUNet (around -0.05dB) and maintaining a signiﬁcant PSNR gap (around +0.5dB) with other deep denoisers like DnCNN and FFDNet.

Original: Note that the time difference between GS-DRUNet and DRUNet light is due to the computation of ∇ g σ via backpropagation.
Revised: Note that the time difference between GS-DRUNet and DRUNet light is due to the computation of ∇ g σ via backpropagation.

Original: These results indicate that GS-DRUNet is likely to yield a competitive and fast PnP algorithm.
Revised: These results indicate that GS-DRUNet is likely to yield a competitive and fast PnP algorithm.

Original: 5. P LUG - AND -P LAY IMAGE RESTORATION
Revised: 5. P LUG - AND -P LAY IMAGE RESTORATION

Original: We apply our PnP algorithm 1 to deblurring in Section 5.2.1, super-resolution in Section 5.2.2 and inpainting in Appendix G.3.
Revised: We apply our PnP Algorithm 1 to deblurring in Section 5.2.1, super-resolution in Section 5.2.2 and inpainting in Appendix G.4.

Original: In all cases, we seek an estimate x ∈ R n of the original clean image ˆ x ∈ R n , from an observation y ∈ R m obtained as y =
Revised: In all cases, we seek an estimate x ∈ R n of the original clean image ˆ x ∈ R n , from an observation y ∈ R m obtained as y =

Original: A ˆ x + ξ ν , with A a m × n degradation matrix and ξ ν a white Gaussian noise with zero mean and standard deviation ν .
Revised: A ˆ x + ξ ν , with A a m × n degradation matrix and ξ ν a white Gaussian noise with zero mean and standard deviation ν .

Original: The objective function minimized by Algorithm 1 is
Revised: The objective function minimized by Algorithm 1 is

Original: [Equation]
Revised: [Equation]

Original: In practice, for ν > 0 , we multiply F in equation 12 by ν 2 and consider F = f
Revised: In practice, for ν > 0 , we multiply F in equation (12) by ν 2 and consider F = f

Original: + λ ν g σ with f ( x ) =
Revised: + λ ν g σ with f ( x ) =

Original: 12 ||
Revised: 12 ||

Original: Ax − y
Revised: Ax − y

Original: || 2 , g σ ( x ) =
Revised: || 2 , g σ ( x ) =

Original: 12 || N σ ( x ) − x || 

Revised: 12 || N σ ( x ) − x ||

Original: and λ ν = λν 2 .
Revised: and λ ν = λν 2 .

Original: With this formulation, the convergences of iterates and objective values are theorically ensured with Theorems 1 and 2.
Revised: With this formulation, the convergences of iterates and objective values are guaranteed by Theorems 1 and 2.

Original: We demonstrate in Appendix G.3 that our framework can be extended to other kinds of objective functions.
Revised: We demonstrate in Appendix G.4 that our framework can be extended to other kinds of objective functions.

Original: For example, inpainting noise-free input images lead to a non differentiable data term f .
Revised: For example, inpainting noise-free input images leads to a non differentiable data-ﬁdelity term f .

Original: Due to the large computational time of some compared methods, we use for evaluation and comparison a subset of 10 color images taken from the CBSD68 dataset (CBSD10) together with the three famous set3C images (butterﬂy, leaves and starﬁsh).
Revised: Due to the large computational time of some compared methods, we use for evaluation and comparison a subset of 10 color images taken from the CBSD68 dataset (CBSD10) together with the three famous set3C images (butterﬂy, leaves and starﬁsh).

Original: Some quantitative results run on the full CBSD68 dataset will alsobe given in Appendix G. All input images are center-cropped to the size 256 × 256 .
Revised: Quantitative results run on the full CBSD dataset are also given in Appendix G. All input images are center-cropped to the size 256 × 256 .

Revised: For each IR problem, we provide default values for the parameters σ and λ that can be used to treat successfully a large class of images and degradations.

Revised: The inﬂuence of both parameters is analyzed in Appendix G.5.

Revised: Performance could be marginally improved by tuning λ for each image, for example with the method of Wei et al.

Revised: based on reinforcement learning.

Original: In all our experiments, backtracking is performed with η = 0 .
Revised: In all our experiments, backtracking is performed with η = 0 .

Original: 9 and γ = 0 .
Revised: 9 and γ = 0 .

Original: We estimated (see Appendix D) that on a majority of images, the Lipschitz constant L of ∇ g σ is slightly larger than 1 .
Revised: We estimated (see Appendix D) that on a majority of images, the Lipschitz constant L of ∇ g σ is slightly larger than 1 .

Original: As convergence is ensured for λ ν τ = ν 2 λτ < L , we choose to set the initial stepsize to τ 0 = ( ν 2 λ ) − 1 .
Revised: As convergence is ensured for λ ν τ = ν 2 λτ < L , we choose to set the initial stepsize to τ 0 = ( ν 2 λ ) − 1 .

Original: At the ﬁrst iteration, the gradient step in equation 9 is thus exactly D σ .
Revised: At the ﬁrst iteration, the gradient step in equation (9) is thus exactly D σ .

Original: We observe that in the majority of our experiments, backtracking is never activated.
Revised: We observe that in the majority of our experiments, backtracking is never activated.

Original: Finally, the algorithm is initialized with a proximal step and terminates when the relative difference between two consecutive values of the objective function is less than (cid:15) or the number of iterations exceeds K .
Revised: Finally, the algorithm is initialized with a proximal step and terminates when the relative difference between two consecutive values of the objective function is less than (cid:15) or the number of iterations exceeds K .

Original: 5.2. D EBLURRING
Revised: 5.2. D EBLURRING

Original: For image deblurring, the degradation operator A
Revised: For image deblurring, the degradation operator A

Original: = H is a convolution performed with circular boundary conditions.
Revised: = H is a convolution performed with circular boundary conditions.

Original: Therefore, H = F ∗ Λ F , where F is the orthogonal matrix of the discrete
Revised: Therefore, H = F ∗ Λ F , where F is the orthogonal matrix of the discrete

Original: Fourier transform (and F ∗ its inverse), and Λ is a diagonal matrix.
Revised: Fourier transform (and F ∗ its inverse), and Λ is a diagonal matrix.

Original: The proximal operator of the data term f ( x ) = 12 || Hx − y
Revised: The proximal operator of the data-ﬁdelity term f ( x ) = 12 || Hx − y

Original: ||
Revised: ||

Original: 2 involves only element-wise inversion and writes
Revised: 2 involves only element-wise inversion and writes

Original: Prox [Equation]
Revised: Prox τf ( z ) =

Revised: F ∗ ( I n + τ Λ ∗ Λ) − 1 F ( τH T y + z ) .

Original: We demonstrate the effectiveness of our method on a large variety of blur kernels (represented in
Revised: We demonstrate the effectiveness of our method on a large variety of blur kernels (represented in

Original: Table 2) and noise levels.
Revised: Table 2) and noise levels.

Original: As in Zhang et al.
Revised: As in Zhang et al.

Original: (2017b)
Revised: (2017b)

Original: ; Pesquet et al.
Revised: ; Pesquet et al.

Original: ; Zhang et al.
Revised: ; Zhang et al.

Original: (2021), we use the 8 real-world camera shake kernels proposed in Levin et al.
Revised: (2021), we use the 8 real-world camera shake kernels proposed in Levin et al.

Original: (2009) as well as the 9 × 9 uniform kernel and the 25 × 25 Gaussian kernel with standard deviation 1 .
Revised: (2009) as well as the 9 × 9 uniform kernel and the 25 × 25 Gaussian kernel with standard deviation 1 .

Original: 6 (as in Romano et al.
Revised: 6 (as in Romano et al.

Original: We consider Gaussian noise with 3 noise levels ν ∈ { 2 . 55 , 7 . 65 , 12 . 75 } / 255 i-e ν ∈ { 0 . 01 , 0 .
Revised: We consider Gaussian noise with 3 noise levels ν ∈ { 2 . 55 , 7 . 65 , 12 . 75 } / 255 i.e. ν ∈ { 0 . 01 , 0 .

Original: For all noise levels, we set σ = 1 .
Revised: For all noise levels, we set σ = 1 .

Original: 8 ν , λ ν = ν 2 λ = 0 .
Revised: 8 ν , λ ν = ν 2 λ = 0 .

Original: 1 for motion blur (kernels (a) to (h)) and λ ν = 0 .
Revised: 1 for motion blur (kernels (a) to (h)) and λ ν = 0 .

Original: 075 for static blur (kernels (i) and (j)).
Revised: 075 for static blur (kernels (i) and (j)).

Original: Initialization is done with z 

Revised: Initialization is done with z

Original: =
Revised: =

Revised: y

Revised: but we show in

Revised: Appendix G.6 the robustness to the initialization.

Original: y and the stopping criteria are (cid:15) =
Revised: The stopping criteria are (cid:15) =

Original: 10 − 5 and K = 400 .
Revised: 10 − 5 and K = 400 .

Original: We compare in Table 2 our method (GS-PnP) against the patch-based method EPLL (Zoran & Weiss, 2011; Hurault et al., 2018), the deep PnP methods IRCNN (Zhang et al., 2017b), DPIR (Zhang et al., 2021), MMO (Pesquet et al., 2021), and the “RED-FP” algorithm (Romano et al., 2017)  referred to as RED.
Revised: We compare in Table 2 our method (GS-PnP) against the patch-based method EPLL (Zoran & Weiss, 2011; Hurault et al., 2018), the deep PnP methods IRCNN (Zhang et al., 2017b), DPIR (Zhang et al., 2021), MMO (Pesquet et al., 2021), and the “RED-FP” algorithm (Romano et al., 2017) (with TNRD denoiser (Chen & Pock, 2016)) referred to as RED.

Original: Both IRCNN and DPIR realize PnP-HQS with a fast decrease of τ and σ in a few iterations (8 iterations for DPIR) without guarantee of convergence.
Revised: Both IRCNN and DPIR use PnP-HQS with a fast decrease of τ and σ in a few iterations (8 iterations for DPIR) without guarantee of convergence.

Original: DPIR uses the DRUNet denoiser from Table 1.
Revised: DPIR uses the DRUNet denoiser from Table 1.

Original: MMO is the only compared method that guarantees convergence by plugging a DnCNN denoiser trained with Lipshitz constraints, but the only pretrained network given is for very low noise level.
Revised: MMO is the only compared method that guarantees convergence by plugging a DnCNN denoiser trained with Lipschitz constraints, but the only pretrained network given is for very low noise level.

Original: Finally, as RED only treats the Y channel in the YCbCr color space, we also indicate in Appendix G, for RED and the proposed method, the PSNR evaluated on the Y channel only.
Revised: Finally, as RED only treats the Y channel in the YCbCr color space, we also indicate in Appendix G.2, for RED and the proposed method, the PSNR evaluated on the Y channel only.

Original: Among all methods, GS-PnP closely follows DPIR in terms of PSNR for low noise level but performs equally or better for higher noise levels.
Revised: Among all methods, GS-PnP closely follows DPIR in terms of PSNR for low noise level but performs equally or better for higher noise levels.

Original: Other comparisons are conducted in Appendix G. on the Set3c dataset (Table 4) and on the full CBSD68 dataset (Table 5).
Revised: Other comparisons are conducted in Appendix G. on the Set3c dataset (Table 4) and on the full CBSD68 dataset (Table 5).

Original: These results all conﬁrm that GS-PnP reaches state-of-the-art in PnP deblurring, for a variety of kernels and noise levels.
Revised: These results all conﬁrm that GS-PnP reaches state-of-the-art in PnP deblurring, for a variety of kernels and noise levels.

Original: [Table]
Revised: [Table]

Original: [Table] [Table]
Revised: [Table] [Table]

Original: [Table] [Table]
Revised: [Table] [Table]

Original: [Table] [Table] 
Revised: [Table] [Table] [Table]

Original: [Table] [Table] 
Revised: [Table] [Table] [Table]

Original: [Table] [Table] 
Revised: [Table] [Table] [Table]

Original: [Table] [Table] 
Revised: [Table] [Table] [Table]

Original: [Table] [Table] 
Revised: [Table] [Table] [Table]

Original: [Table] 
Revised: [Table] [Table] [Table] [Table]

Original: [Table] 
Revised: [Table] [Table] [Table] [Table]

Original: [Table] 
Revised: [Table] [Table] [Table] [Table]

Original: [Table] 
Revised: [Table] [Table] [Table] [Table]

Original: [Table] 
Revised: [Table] [Table] [Table] [Table]

Original: [Table] 
Revised: [Table] [Table] [Table] [Table]

Original: [Table] 
Revised: [Table] [Table] [Table] [Table]

Original: [Table] [Table] 
Revised: [Table] [Table] [Table]

Original: [Table][Table] [Table] 16 32.50 GS
Revised: [Table]

Original: [Table]
Revised: [Table]

Original: PnP

Original: [Table]

Original: [Table]

Original: [Table]

Original: EPLL

Original: [Table]
Revised: [Table]

Original: 43 25.16 RED

Original: [Table]

Original: 54 25.26 IRCNN

Original: [Table]

Original: [Table]

Original: 64 28.58 DPIR

Original: [Table]
Revised: [Table]

Original: 81 28.91 GS

Original: [Table]
Revised: [Table]

Original: PnP

Original: [Table]

Original: [Table]

Original: [-EPLL 24 .23 .24 .22 .25 .23 .
-]
Revised: [-EPLL 24 .23 .24 .22 .25 .23 .
-]

Original: [-.23 .24 . 78 24.04 RED 22 .22 .23 .
-]
Revised: [-.23 .24 . 78 24.04 RED 22 .22 .23 .
-]

Original: 78 22.84 IRCNN

Original: [Table]
Revised: [Table]

Original: 83 26.94 DPIR

Original: [Table]
Revised: [Table]

Original: 11 27.42 GS-PnP

Original: [Table] [Table] For qualitative comparison, we show in Figure 1(c-f) the deblurring obtained with various methods on the image “starﬁsh” (from set3C).
Revised: For qualitative comparison, we show in Figure 1(c-f) the deblurring obtained with various methods on the image “starﬁsh” (from set3C).

Original: Note that our algorithm, compared to competing methods, can recover the sharpest edges.
Revised: Note that our algorithm, compared to competing methods, can recover the sharpest edges.

Original: We also give convergence curves that empirically conﬁrm the convergence of the values F ( x k ) (g) and of the residual min 0 ≤ i ≤ k
Revised: We also give convergence curves that empirically conﬁrm the convergence of the values F ( x k ) (g) and of the residual min 0 ≤ i ≤ k

Original: || x
Revised: || x

Original: i +1 − x i
Revised: i +1 − x i

Original: || 2 /
Revised: || 2 /

Original: ||
Revised: ||

Original: x 0 || 2 (h).
Revised: x 0 || 2 (h).

Original: These observations are supported by the additional experiment shown in Appendix G.1, Figure 5.
Revised: These observations are supported by the additional experiment shown in Appendix G.2, Figure 6.

Original: [Figure]
Revised: [Figure]

Original: 5.2. S UPER - RESOLUTION
Revised: 5.2. S UPER - RESOLUTION

Original: For single image super-resolution, the low-resolution image y ∈ R m is obtained from the highresolution one x ∈ R n via y =
Revised: For single image super-resolution, the low-resolution image y ∈ R m is obtained from the highresolution one x ∈ R n via y =

Original: SHx + ξ ν where H ∈ R n × n is the convolution with anti-aliasing kernel.
Revised: SHx + ξ ν where H ∈ R n × n is the convolution with anti-aliasing kernel.

Original: The matrix S is the standard s-fold downsampling matrix of size m × n and n = s 2 × m .
Revised: The matrix S is the standard s-fold downsampling matrix of size m × n and n = s 2 × m .

Original: In this context, we make use of the closed-form calculation of the proximal map for the data term f ( x ) = 12 || SHx − y || 2 , given by Zhao et al.
Revised: In this context, we make use of the closed-form calculation of the proximal map for the data-ﬁdelity term f ( x ) = 12 || SHx − y || 2 , given by Zhao et al.

Original: [Equation]
Revised: [Equation]

Original: (14) where ˆ z τ = τH T S
Revised: (14) where ˆ z τ = τH T S

Original: T
Revised: T

Original: y
Revised: y

Original: + z and Λ =
Revised: + z and Λ =

Original: [Λ 1 , . .
Revised: [Λ 1 , . .

Original: , Λ s 2 ] ∈ R m × n , with Λ = diag(Λ 1 , . .
Revised: , Λ s 2 ] ∈ R m × n , with Λ = diag(Λ 1 , . .

Original: , Λ s 2 ) a blockdiagonal decomposition according to a s × s paving of the Fourier domain.
Revised: , Λ s 2 ) a blockdiagonal decomposition according to a s × s paving of the Fourier domain.

Original: Note that I m + τd ΛΛ ∗ is a m × m diagonal matrix and its inverse is computed element-wise.
Revised: Note that I m + τd ΛΛ ∗ is a m × m diagonal matrix and its inverse is computed element-wise.

Original: As expected, with s = 1 , equation (14) comes down to equation (13).
Revised: As expected, with s = 1 , equation (14) comes down to equation (13).

Original: As in Zhang et al.
Revised: As in Zhang et al.

Original: (2021), we evaluate super-resolution performance on 8 Gaussian blur kernels represented in Table 3: 4 isotropic kernels with different standard deviations ( 0 . 7 , 1 . 2 , 1 . 6 and 2 . 0 ) and 4 anisotropic kernels.
Revised: (2021), we evaluate super-resolution performance on 8 Gaussian blur kernels represented in Table 3: 4 isotropic kernels with different standard deviations ( 0 . 7 , 1 . 2 , 1 . 6 and 2 . 0 ) and 4 anisotropic kernels.

Original: Results are averaged between isotropic and anisotropic.
Revised: Results are averaged between isotropic and anisotropic.

Original: We consider downsampled images at scale s = 2 and s = 3 and Gaussian noise with 3 different noise levels ν ∈ { 0 . 01 , 0 .
Revised: We consider downsampled images at scale s = 2 and s = 3 and Gaussian noise with 3 different noise levels ν ∈ { 0 . 01 , 0 .

Original: Our method (GS-PnP) is compared against bicubic upsampling, RED,
Revised: Our method (GS-PnP) is compared against bicubic upsampling, RED,

Original: IRCNN (“IRCNN+” from Zhang et al.
Revised: IRCNN (“IRCNN+” from Zhang et al.

Original: (2021)) and DPIR.
Revised: (2021)) and DPIR.

Original: We give again in Appendix G.2 the results obtained on the Set3C dataset (Table 7).
Revised: We give again in Appendix G.3 the results obtained on the Set3C dataset (Table 7).

Original: All our results are obtained with λ
Revised: All our results are obtained with λ

Original: ν = ν 2 λ = 0 .
Revised: ν = ν 2 λ = 0 .

Original: 065 and σ = 2 ν .
Revised: 065 and σ = 2 ν .

Original: Initialization z 0 is done with a bicubic interpolation of y (with a shift correction (Zhang et al., 2021)) and the stopping criteria are (cid:15)
Revised: Initialization z 0 is done with a bicubic interpolation of y (with a shift correction (Zhang et al., 2021)) and the stopping criteria are (cid:15)

Original: = 10 − 6 and K = 400 .
Revised: = 10 − 6 and K = 400 .

Original: Our method outperforms in PSNR all other PnP methods over the considered range of blur kernels, noise levels and scale factors.
Revised: Our method outperforms in PSNR all other PnP methods over the considered range of blur kernels, noise levels and scale factors.

Original: We show in Figure 2 the super-resolution of the image “leaves” downsampled by 2, with an isotropic kernel and noise level ν = 0 .
Revised: We show in Figure 2 the super-resolution of the image “leaves” downsampled by 2, with an isotropic kernel and noise level ν = 0 .

Original: GS-PnP (f) recovers more accurately structures and color details than competing approaches (c-e), while converging in terms of function values (g) and residual (h).
Revised: GS-PnP (f) recovers more accurately structures and color details than competing approaches (c-e), while converging in terms of function values (g) and residual (h).

Original: Additional visual comparisons are presented in Appendix G.2.
Revised: Additional visual comparisons are presented in Appendix G.3.

Original: [Table]
Revised: [Table]

Original: [Figure] 
Revised: [Figure] [Table] [Figure] [Table]

Original: [Table] [Figure] [Table]
Revised: [Figure] [Table]

Original: [Figure]
Revised: [Figure]

Original: [Table][Figure] [Table]
Revised: [Table]

Original: [Figure]
Revised: [Figure]

Original: [Table]
Revised: [Table]

Original: [Figure]
Revised: [Figure]

Original: [Table]
Revised: [Table]

Original: [Figure]
Revised: [Figure]

Original: : Super-resolution with various methods on “leaves” (set3C) downsampled by 2 , with the indicated blur kernel and input noise level ν = 0 .
Revised: Super-resolution with various methods on “leaves” (set3C) downsampled by 2 , with the indicated blur kernel and input noise level ν = 0 .

Original: Note that our algorithm is the one that recovers sharpest leaves.
Revised: Note that our algorithm is the one that recovers sharpest leaves.

Original: We empirically verify convergence offunctional values F ( x k ) andresidual γ k = min 0 ≤ i ≤ k || x
Revised: In (g) and (h), we show the evolution of F ( x k ) and γ k = min 0 ≤ i ≤ k || x

Original: i +1 − x i
Revised: i +1 − x i

Original: || 2 /
Revised: || 2 /

Original: ||
Revised: ||

Original: x 0 || 2 .
Revised: x 0 || 2 and in AppendixG.1 the evolution of the PSNR..

Original: The empirical convergence rate is faster than the O ( 1 k ) theoretical worst case rate established in Remark 2.
Revised: The empirical convergence rate is faster than the O ( 1 k ) theoretical worst case rate established in Remark 2.

Original:  C ONCLUSION
Revised:  C ONCLUSION

Original: In this work, we introduce a new PnP algorithm with convergence guarantees.
Revised: In this work, we introduce a new PnP algorithm with convergence guarantees.

Original: A denoiser is trained to realize an exact gradient step on a regularization function that is formulated through a neural network.
Revised: A denoiser is trained to realize an exact gradient step on a regularization function that is formulated through a neural network.

Original: This denoiser is plugged in an iterative scheme closely related to PnP-HQS, which is proved to converge towards a stationary point of an explicit functional.
Revised: This denoiser is plugged in an iterative scheme closely related to PnP-HQS, which is proved to converge towards a stationary point of an explicit functional.

Original: One strength of this approach is to simultaneously allow for a non strongly convex (and not smooth) data term with a denoiser that may not be non-expansive.
Revised: One strength of this approach is to simultaneously allow for a non strongly convex (and not smooth) data-ﬁdelity term with a denoiser that may not be non-expansive.

Original: Experiments conducted on ill-posed imaging problems (deblurring, superresolution, inpainting) conﬁrm the convergence results and show that the proposed PnP algorithm reaches state-of-the-art image restoration performance.
Revised: Experiments conducted on ill-posed imaging problems (deblurring, super-resolution, inpainting) conﬁrm the convergence results and show that the proposed PnP algorithm reaches state-of-the-art image restoration performance.

Original: This work also opens several research perspectives.
Revised: This work also opens several research perspectives.

Original: One could ﬁrst examine which information is encoded in the proposed prior.
Revised: One could ﬁrst examine which information is encoded in the proposed prior.

Original: For example, based on the sharp visual results, one can question if a relation can be drawn between this prior and the gradient energy or sparsity.
Revised: For example, based on the sharp visual results, one can question if a relation can be drawn between this prior and the gradient energy or sparsity.

Original: Also, it would be interesting to see if the recovery analysis developed in Liu et al.
Revised: Also, it would be interesting to see if the recovery analysis developed in Liu et al.

Original: (2021) can adapt to the proposed framework.
Revised: (2021) can adapt to the proposed framework.

Original:  R EPRODUCIBILITY S TATEMENT
Revised:  R EPRODUCIBILITY S TATEMENT

Original: Anonymous source code is given in supplementary material.
Revised: Anonymous source code is given in supplementary material.

Original: It contains a README.md ﬁle that explains step by step how to run the algorithm and replicate the results of the paper.
Revised: It contains a README.md ﬁle that explains step by step how to run the algorithm and replicate the results of the paper.

Original: Moreover, the pseudocode of our algorithm is given Algorithm 1.
Revised: Moreover, the pseudocode of our algorithm is given Algorithm 1.

Original: In Section 5 it is precisely detailed how all the hyper-parameters are chosen and, for each experiment, which dataset is used.
Revised: In Section 5 it is precisely detailed how all the hyper-parameters are chosen and, for each experiment, which dataset is used.

Original: As for the theoretical results presented in Section 4, complete proofs are given in the appendixes.
Revised: As for the theoretical results presented in Section 4, complete proofs are given in the appendixes.

Original:  E THICS S TATEMENT
Revised:  E THICS S TATEMENT

Original: We believe that this work does not raise potential ethical concerns.
Revised: We believe that this work does not raise potential ethical concerns.

Original: A P ROOF OF T HEOREM
Revised: A P ROOF OF T HEOREM

Original: We ﬁrst remind that a function f : R n −→ R ∪ + ∞ is proper if its domaindom ( f ) = { x ∈ R , f ( x ) <
Revised: We ﬁrst remind that a function f : R n −→ R ∪ + ∞ is proper if its domain

Original: + ∞} (15) is non empty.
Revised: [Equation] 15) is non empty.

Original: Also, recall that f is lower semicontinuous at x ∗ if lim inf x → x ∗ f ( x ) ≥ f ( x ∗ ) .
Revised: Also, recall that f is lower semicontinuous at x ∗ if lim inf x → x ∗ f ( x ) ≥ f ( x ∗ ) .

Original: Proof.
Revised: Proof.

Original: (i) For ease of notation, we consider λ = 1 .
Revised: (i) For ease of notation, we consider λ = 1 .

Original: The generalisation for any λ > 0 is straightforward by rescaling g σ (and L ) accordingly.
Revised: The generalisation for any λ > 0 is straightforward by rescaling g σ (and L ) accordingly.

Original: We denote the proximal gradient ﬁxed point operator T τ =
Revised: We denote the proximal gradient ﬁxed point operator T τ =

Original: Prox τf ◦ (Id − τ ∇ x g σ ) , the objective function F = f + g σ
Revised: Prox τf ◦ (Id − τ ∇ x g σ ) , the objective function F = f + g σ

Original: and we introduce
Revised: and we introduce

Original: [Equation]
Revised: [Equation]

Original: We have
Revised: We have

Original: [Equation]
Revised: [Equation]

Original: By deﬁnition for the arg min , x k +1 =
Revised: By deﬁnition for the arg min , x k +1 =

Original: T τ ( x k ) ⇒ Q
Revised: T τ ( x k ) ⇒ Q

Original: τ ( x k +1 , x k ) ≤ Q τ ( x k , x k ) .
Revised: τ ( x k +1 , x k ) ≤ Q τ ( x k , x k ) .

Original: Moreover, with g σ being L -smooth, we have by the descent lemma, for any τ ≤ 1 L and any x, y ∈ R n ,
Revised: Moreover, with g σ being L -smooth, we have by the descent lemma, for any τ ≤ 1 L and any x, y ∈ R n ,

Original: [Equation]
Revised: [Equation]

Revised: so that for every x, y ∈ R n ,

Revised: Q τ

Original: [Equation]
Revised: [Equation]

Original: so that for every x, y ∈ R n ,
Revised: x, x ) = F ( x ) and Q τ ( x, y ) ≥

Original: [Equation]
Revised: [Equation]

Original: Therefore, at iteration k ,
Revised: Therefore, at iteration k ,

Original: [Equation]
Revised: [Equation]

Original: 20) ( F ( x k )) is thus non-increasing.
Revised: [Equation] ( F ( x k )) is thus non-increasing.

Original: Since F is lower-bounded, ( F ( x k )) thus converges to a limit F ∗ .
Revised: Since F is lower-bounded, ( F ( x k )) thus converges to a limit F ∗ .

Original: (ii) Note that Q τ ( x k +1 , x k ) ≤ Q τ ( x k , x k ) implies
Revised: (ii) Note that Q τ ( x k +1 , x k ) ≤ Q τ ( x k , x k ) implies

Original: [Equation]
Revised: [Equation]

Original: Using also (18) with stepsize 1 L , we get
Revised: Using also (18) with stepsize 1 L , we get

Original: [Equation]
Revised: [Equation]

Original: Summing over k = 0 , 1 , ..., m gives 
Revised: Summing over k = 0 , 1 , ..., m gives m (cid:88)

Original: [Equation]
Revised: [Equation]

Original: Therefore, lim k →∞ || x k +1 − x k ||
Revised: Therefore, lim k →∞ || x k +1 − x k ||

Original: = 0 .
Revised: = 0 .

Original: (iii)
Revised: (iii)

Original: We begin by the two following lemmas characterizing the proximal gradient descent operator
Revised: We begin by the two following lemmas characterizing the proximal gradient descent operator

Original: T τ =
Revised: T τ =

Original: Prox τf ◦ (Id − τ ∇ x g σ ) .
Revised: Prox τf ◦ (Id − τ ∇ x g σ ) .

Original: Lemma 1.
Revised: Lemma 1.

Original: With the assumptions of Theorem 1, for x ∗ ∈ R n , x ∗ is a ﬁxed point of the proximal gradient descent operator T τ =
Revised: With the assumptions of Theorem 1, for x ∗ ∈ R n , x ∗ is a ﬁxed point of the proximal gradient descent operator T τ =

Original: Prox τf ◦ (Id − τ ∇ x g σ ) , i.e. T τ ( x ∗ ) =
Revised: Prox τf ◦ (Id − τ ∇ x g σ ) , i.e. T τ ( x ∗ ) =

Original: x ∗ , if and only if x ∗ is a stationary point of problem (10), i.e. −∇ g σ ( x ∗ )
Revised: x ∗ , if and only if x ∗ is a stationary point of problem (10), i.e. −∇ g σ ( x ∗ )

Original: ∈ ∂f ( x ∗ ) .
Revised: ∈ ∂f ( x ∗ ) .

Original: Proof.
Revised: Proof.

Original: By deﬁnition of the proximal operator, we have
Revised: By deﬁnition of the proximal operator, we have

Original: [Equation]
Revised: [Equation]

Original: [Equation]

Original: Lemma 2.
Revised: Lemma 2.

Original: With the assumptions of Theorem 1, T τ is 1 + τL Lipschitz.
Revised: With the assumptions of Theorem 1, T τ is 1 + τL Lipschitz.

Original: Proof.
Revised: Proof.

Original: Using the fact that for f convex, Prox τf is 1-Lipschitz ((Bauschke & Combettes, 2011,
Revised: Using the fact that for f convex, Prox τf is 1-Lipschitz ((Bauschke & Combettes, 2011,

Original: Proposition 12.28)), and by the Lipschitz property of ∇ x g σ
Revised: Proposition 12.28)), and by the Lipschitz property of ∇ x g σ

Original: [Equation]
Revised: [Equation]

Original: Note that, by nonconvexity of g σ , the ﬁxed point operator T τ is not necessarily non-expansive, but we can still show the convergence of the ﬁxed-point algorithm towards a critical point of the objective function.
Revised: Note that, by nonconvexity of g σ , the ﬁxed point operator T τ is not necessarily non-expansive, but we can still show the convergence of the ﬁxed-point algorithm towards a critical point of the objective function.

Original: We can now turn to the proof of (iii).
Revised: We can now turn to the proof of (iii).

Original: Let x ∗ be a cluster point of ( x k ) k ≥ 0 .
Revised: Let x ∗ be a cluster point of ( x k ) k ≥ 0 .

Original: Then there exists a subsequence ( x k j ) j ≥ 0 converging to x ∗ .
Revised: Then there exists a subsequence ( x k j ) j ≥ 0 converging to x ∗ .

Original: We have ∀ j ≥ 0 ,
Revised: We have ∀ j ≥ 0 ,

Original: [Equation]
Revised: [Equation]

Original:  Using (ii), the right-hand side of the inequity tends to 0 as j → ∞ .
Revised: [Equation] [Equation] Using (ii), the right-hand side of the inequity tends to 0 as j → ∞ .

Original: Thus || x ∗ − T τ ( x ∗ ) ||
Revised: Thus || x ∗ − T τ ( x ∗ ) ||

Original: = 0 and x ∗ =
Revised: = 0 and x ∗ =

Original: T τ ( x ∗ ) , which by Lemma 1 means that x ∗ is a stationary point of problem (10).
Revised: T τ ( x ∗ ) , which by Lemma 1 means that x ∗ is a stationary point of problem (10).

Original: B O N THE BOUNDEDNESS OF ( x k )
Revised: B O N THE BOUNDEDNESS OF ( x k )

Original: In order to obtain convergence of the iterates, in Theorem 2 the generated sequence ( x k ) is assumed to be bounded.
Revised: In order to obtain convergence of the iterates, in Theorem 2 the generated sequence ( x k ) is assumed to be bounded.

Original: In the experiments (Section 5), we observed that under the rest of assumptions of
Revised: In the experiments (Section 5), we observed that under the rest of assumptions of

Original: Theorem 2, boundedness was always veriﬁed.
Revised: Theorem 2, boundedness was always veriﬁed.

Original: A sufﬁcient condition for the boundedness of the iterates is the coercivity of the objective function, that is, lim | x |→∞ F ( x ) =
Revised: A sufﬁcient condition for the boundedness of the iterates is the coercivity of the objective function, that is, lim | x |→∞ F ( x ) =

Original: + ∞ (because the non-increasing property gives F ( x k ) ≤ F ( x 0 ) ).
Revised: + ∞ (because the non-increasing property gives F ( x k ) ≤ F ( x 0 ) ).

Original: Similar to Laumont et al.
Revised: Similar to Laumont et al.

Original: (2021), we can constrain F to be coercive by choosing a convex compact set C ⊂ R n where the iterates should stay and by adding an extra term to the regularization g σ :
Revised: (2021), we can constrain F to be coercive by choosing a convex compact set C ⊂ R n where the iterates should stay and by adding an extra term to the regularization g σ :

Original: [Equation]
Revised: [Equation]

Original: with Π C the Euclidian projection on C .
Revised: with Π C the Euclidian projection on C .

Original: As g σ is differentiable, the gradient step becomes
Revised: As g σ is differentiable, the gradient step becomes

Original: [Equation]
Revised: [Equation]

Original: In our experiments, we choose the compact set C as C =
Revised: In our experiments, we choose the compact set C as C =

Original: [ − 1 , 2] n .
Revised: [ − 1 , 2] n .

Original: In practice we observe that all the iterates always remain in C and that the extra regularization term || x − Π C ( x ) || 2 is never activated.
Revised: In practice we observe that all the iterates always remain in C and that the extra regularization term || x − Π C ( x ) || 2 is never activated.

Original: Therefore, we don’t present this technical adaptation in Algorithm 1 but we let the reader aware that boundedness of ( x k ) is not a limiting assumption.
Revised: Therefore, we don’t present this technical adaptation in Algorithm 1 but we let the reader aware that boundedness of ( x k ) is not a limiting assumption.

Original: C KL PROPERTY
Revised: C KL PROPERTY

Original: Deﬁnition 1.
Revised: Deﬁnition 1.

Original: Kurdyka-Lojasiewicz (KL) property (taken from Attouch et al. (2010))
Revised: Kurdyka-Lojasiewicz (KL) property (taken from Attouch et al. (2010))

Original: (a A function f : R n −→ R ∪ + ∞ is said to have the Kurdyka-Lojasiewicz property at 
Revised: (a) A function f : R n −→ R ∪ + ∞ is said to have the Kurdyka-Lojasiewicz property at x ∗ ∈ dom ( f ) if there exists η ∈ (0 , + ∞ ) , a neighborhood U of x ∗ and a continuous concave function ψ : [0 , η ) −→ R + such that ψ (0) = 0 , ψ is C 1 on (0 , η ) , ψ (cid:48) > 0 on ( η )

Original:  x∗ ∈ dom (  f  )if there exists η
Revised: and ∀ x ∈ U ∩ [ f ( x ∗ ) < f < f ( x ∗ )

Original: ∈
Revised: + η ]

Original: 0 , + ∞ ) , a neighborhood U of x ∗ and a continuous concave function ψ : [0 , η ) −→ R + such that ψ (0) = 0 , ψ is C 1 on (0 , η ) , ψ (cid:48) > 0 on (0 , η ) and ∀ x ∈ U ∩ [ f ( x ∗ ) < f < f ( x ∗ ) + η ] , the Kurdyka-Lojasiewicz inequality holds :
Revised: , the Kurdyka -Lojasiewicz inequality holds :

Original: [Equation]
Revised: [Equation]

Original: (b) Proper lower semicontinuous functions which satisfy the Kurdyka-Lojasiewicz inequality at each point of dom∂f are called KL functions.
Revised: (b) Proper lower semicontinuous functions which satisfy the Kurdyka-Lojasiewicz inequality at each point of dom∂f are called KL functions.

Original: This condition can be interpreted as the fact that, up to a reparameterization, the function is sharp i.e. we can bound its subgradients away from 0.
Revised: This condition can be interpreted as the fact that, up to a reparameterization, the function is sharp i.e. we can bound its subgradients away from 0.

Original: A big class of functions that have the KL-property is given by real semi-algebraic functions.
Revised: A big class of functions that have the KL-property is given by real semi-algebraic functions.

Original: For more details and interpretations, we refer to Attouch et al.
Revised: For more details and interpretations, we refer to Attouch et al.

Original: (2010) and Bolte et al.
Revised: (2010) and Bolte et al.

Original: D L IPSCHITZ CONSTANT OF ∇ g σ
Revised: D L IPSCHITZ CONSTANT OF ∇ g σ

Original: First, let us give a result which ensures that a large class of neural networks trained with differentiable activation functions have Lipschitz gradients with respect to the input image.
Revised: First, let us give a result which ensures that a large class of neural networks trained with differentiable activation functions have Lipschitz gradients with respect to the input image.

Original: Proposition 2.
Revised: Proposition 2.

Original: Let H =
Revised: Let H =

Original: h p ◦ . . .
Revised: h p ◦ . . .

Original: ◦
Revised: ◦

Original: h 1 be a composition of differentiable functions
Revised: h 1 be a composition of differentiable functions

Original: h i : R d i − 1 → R d i .
Revised: h i : R d i − 1 → R d i .

Original: Let us assume
Revised: Let us assume

Original: that for any i the differential map h (cid:48)
Revised: that for any i the differential map h (cid:48)

Original: i is bounded and Lipschitz.
Revised: i is bounded and Lipschitz.

Original: Then H (cid:48) is Lipschitz.
Revised: Then H (cid:48) is Lipschitz.

Original: Proof.
Revised: Proof.

Original: Let us denote H
Revised: Let us denote H

Original: i
Revised: i

Original: = h
Revised: = h

Original: i ◦ . . .
Revised: i ◦ . . .

Original: ◦
Revised: ◦

Original: h 1 (and by convention, H 

Revised: h 1 (and by convention, H

Original: = Id ).
Revised: = Id ).

Original: Let (cid:107) h (cid:48)
Revised: Let (cid:107) h (cid:48)

Original: i (cid:107) ∞ be the best uniform bound on the operator norms (cid:107) h (cid:48)
Revised: i (cid:107) ∞ be the best uniform bound on the operator norms (cid:107) h (cid:48)

Original: i ( x ) (cid:107) , x ∈ R d i − 1 (which is also the best Lipschitz constant of h i ).
Revised: i ( x ) (cid:107) , x ∈ R d i − 1 (which is also the best Lipschitz constant of h i ).

Original: Let us also denote (cid:107) h (cid:48)
Revised: Let us also denote (cid:107) h (cid:48)

Original: i (cid:107) Lip the Lipschitz constant of h (cid:48)
Revised: i (cid:107) Lip the Lipschitz constant of h (cid:48)

Original: i .
Revised: i .

Original: The chain rule gives that for any x ,
Revised: The chain rule gives that for any x ,

Original: H (cid:48)
Revised: H (cid:48)

Original: ( x ) can be expressed as a composition of linear maps
Revised: ( x ) can be expressed as a composition of linear maps

Original: [Equation]
Revised: [Equation]

Original: Therefore, for any x, y ,
Revised: Therefore, for any x, y ,

Original: [Equation]
Revised: [Equation]

Original: We can thus bound the operator norms
Revised: We can thus bound the operator norms

Original: [Equation]
Revised: [Equation]

Original:  and thus 
Revised: [Equation] 34) and thus (cid:107) H

Original: [Equation]
Revised: [Equation]

Original: 35) which concludes because the chain-rule ensures that (cid:107) H(cid:48) i (cid:107) ∞ ≤
Revised: H

Original: [Equation]
Revised: [Equation]

Original: h
Revised: (cid:107)

Original: [Equation]
Revised: [Equation]

Original:  i 
Revised: 35) which concludes because the chain-rule ensures that (cid:107) H (cid:48) i (cid:107) ∞ ≤ (cid:107) h (cid:48)

Original: [Equation] ∞ . . .
Revised: i (cid:107) ∞ . . .

Original: (cid:107) h (cid:48) 1 (cid:107) ∞ .
Revised: (cid:107) h (cid:48) 1 (cid:107) ∞ .

Original: Proposition 2 applies for a neural network obtained as a composition of fully-connected layers with
Revised: Proposition 2 applies for a neural network obtained as a composition of fully-connected layers with

Original: ELU activation functions, that is, by composing functions of the form
Revised: ELU activation functions, that is, by composing functions of the form

Original: [Equation]
Revised: [Equation]

Original: 36) where A is a matrix, b a vector and E is the element-wise ELU deﬁned by
Revised: where A is a matrix, b a vector and E is the element-wise ELU deﬁned by

Original: [Equation]
Revised: [Equation]

Original: It is easy to see that E is differentiable and that E (cid:48) is 1 -Lipschitz
Revised: It is easy to see that E is differentiable and that E (cid:48) is 1 -Lipschitz

Original: and (cid:107) E (cid:48)
Revised: and (cid:107) E (cid:48)

Original: (cid:107) ∞ ≤ 1 .
Revised: (cid:107) ∞ ≤ 1 .

Original: Thereforeh (cid:48) [Equation] x [Equation]
Revised: Therefore

Original: =

Original: E

Original: [Equation]
Revised: [Equation]

Original: is also bounded and Lipschitz.
Revised: is also bounded and Lipschitz.

Original: Let us also mention that this proposition encompasses the case of U-nets which, in addition to composing fully-connected layers, also integrates skip-connections.
Revised: Let us also mention that this proposition encompasses the case of U-nets which, in addition to composing fully-connected layers, also integrates skip-connections.

Original: For example, taking a skipconnection on a composition h 3 ◦ h 2 ◦ h 1 amounts to deﬁning
Revised: For example, taking a skipconnection on a composition h 3 ◦ h 2 ◦ h 1 amounts to deﬁning

Original: [Equation]
Revised: [Equation]

Original: This can be simply rewritten H =
Revised: This can be simply rewritten H =

Original: h 3 ◦ ˜ h
Revised: h 3 ◦ ˜ h

Original: 2 ◦ h 1 where
Revised: 2 ◦ h 1 where

Original: [Equation]
Revised: [Equation]

Original: It is then clear that ˜ h 2 has bounded Lipschitz differential as soon as h 1 and h 2 do.
Revised: It is then clear that ˜ h 2 has bounded Lipschitz differential as soon as h 1 and h 2 do.

Original: The bound obtained in the proof of Proposition 2 is exponential in the depth of the neural network.
Revised: The bound obtained in the proof of Proposition 2 is exponential in the depth of the neural network.

Original: We now provide some experiments showing that, in practice, the Lipschitz constant of ∇ g σ does not explode.
Revised: We now provide some experiments showing that, in practice, the Lipschitz constant of ∇ g σ does not explode.

Original: With the parameterization proposed in equation 5, g σ is twice differentiable with the right choice of the activation functions in the CNN N σ ( e.g. ELU).
Revised: With the parameterization proposed in equation (5), g σ is twice differentiable with the right choice of the activation functions in the CNN N σ ( e.g. ELU).

Original: We show in Figure 3, for various noise levels σ , the distribution of ||∇ 2 g σ ( x ) || S on the training image set X , estimated with power iterations.
Revised: We show in Figure 3, for various noise levels σ , the distribution of ||∇ 2 g σ ( x ) || S on the training image set X , estimated with power iterations.

Original: The computed value varies a lot across images.
Revised: The computed value varies a lot across images.

Original: Hence approximating the Lipschitz constant of ∇ g σ with L = max x ∈ X ||∇ 2 g σ ( x ) || S would lead to under-estimated stepsizes and slow convergence on most images.
Revised: Hence approximating the Lipschitz constant of ∇ g σ with L = max x ∈ X ||∇ 2 g σ ( x ) || S would lead to under-estimated stepsizes and slow convergence on most images.

Original: Backtracking solves this issue by ﬁnding at each iteration the optimal stepsize allowing sufﬁcient decrease of the objective function.
Revised: Backtracking solves this issue by ﬁnding at each iteration the optimal stepsize allowing sufﬁcient decrease of the objective function.

Original: E P ROOF OF P ROPOSITION
Revised: E P ROOF OF P ROPOSITION

Original: Proof.
Revised: Proof.

Original: For a given stepsize τ , we showed in Appendix A, equation (22) that
Revised: For a given stepsize τ , we showed in Appendix A, equation (22) that

Original: [Equation]
Revised: [Equation]

Original: Taking τ < 1 − 2 γ L , we get 12 ( 1 τ − L ) > γτ
Revised: Taking τ < 1 − 2 γ L , we get 12 ( 1 τ − L ) > γτ

Original: so that
Revised: so that

Original: [Equation]
Revised: [Equation]

Original: Hence, when τ < 1 − 2 γ L , the sufﬁcient decrease condition equation 42 is satisﬁed and the backtracking procedure ( τ ←− ητ ) must end.
Revised: Hence, when τ < 1 − 2 γ L , the sufﬁcient decrease condition equation (42) is satisﬁed and the backtracking procedure ( τ ←− ητ ) must end.

Original: In the proof of Theorem 1, we can replace the sufﬁcient decrease (22) by (42) and ﬁnish the proof with the same arguments.
Revised: In the proof of Theorem 1, we can replace the sufﬁcient decrease (22) by (42) and ﬁnish the proof with the same arguments.

Original: In the same way, in the proof of Theorem 2 given in Attouch et al.
Revised: In the same way, in the proof of Theorem 2 given in Attouch et al.

Original: (2013)[Theorem 5.1], our sufﬁcient decrease (42) replaces Attouch et al.
Revised: (2013)[Theorem 5.1], our sufﬁcient decrease (42) replaces Attouch et al.

Original: (2013, Equation (52)).
Revised: (2013, Equation (52)).

Original: F E XPANSIVENESS OF THE DENOISER
Revised: F E XPANSIVENESS OF THE DENOISER

Original: As g σ is not necessarily convex, our GS-DRUNet denoiser D σ = Id −∇ g σ is not necessarily nonexpansive and neither is the gradient step Id − λτ ∇ g σ .
Revised: As g σ is not necessarily convex, our GS-DRUNet denoiser D σ = Id −∇ g σ is not necessarily nonexpansive and neither is the gradient step Id − λτ ∇ g σ .

Original: This is not an issue as, unlike previous theoretical PnP studies (Terris et al., 2020; Reehorst & Schniter, 2018), our convergence results do not require a non-expansive denoising step.
Revised: This is not an issue as, unlike previous theoretical PnP studies (Terris et al., 2020; Reehorst & Schniter, 2018), our convergence results do not require a non-expansive denoising step.

Original: To advocate that our method converges without this assumption, we show in Figure 4 the evolution of 
Revised: To advocate that our method converges without this assumption, we show in Figure 4 the evolution of || D σ ( x k +1 ) − D σ ( x k )

Original: [Figure]
Revised: || ||

Original:  along the algorithm that was run to obtain the super-resolution results of Figure 6.
Revised: x k +1 − x k || along the algorithm that was run to obtain the super-resolution results of Figure 7.

Original: In this experiment, backtracking did not get activated and stayed ﬁxed at λτ = 1 .
Revised: In this experiment, backtracking did not get activated and stayed ﬁxed at λτ = 1 .

Original: The gradient step in the PGD algorithm was thus simply a denoising step D σ = Id − λτ ∇ g σ .
Revised: The gradient step in the PGD algorithm was thus simply a denoising step D σ = Id − λτ ∇ g σ .

Original: Note that the Lipschitz constant of D σ goes above 1 but convergence is still observed as shown by the two convergence curves in Figure 6.
Revised: Note that the Lipschitz constant of D σ goes above 1 but convergence is still observed as shown by the two convergence curves in Figure 7.

Original: G A DDITIONAL E XPERIMENTS G. D EBLURRING
Revised: G A DDITIONAL E XPERIMENTS G. PSNR CONVERGENCE CURVES

Revised: We ﬁrst plot Figure 5 the evolution of the PSNR along the iterations of the PnP algorithm during the experiments of Figure 1 and Figure 2.

Revised: This illustrates that the minimization of F coincides with the maximization of the PSNR, which supports the interest of the optimized functional F = f + λg σ .

Revised: [Figure]

Revised: G. D EBLURRING

Original: We give here additional image deblurring experiments.
Revised: We give here additional image deblurring experiments.

Original: We ﬁrst present the PSNR performance comparison on the Set3c dataset in Table 4.
Revised: We ﬁrst present the PSNR performance comparison on the Set3c dataset in Table 4.

Original: We also provide an evaluation of the 3 best methods (GS-PnP, DPIR and IRCNN) on the full CBSD68 dataset in Table 5.
Revised: We also provide an evaluation of the 3 best methods (GS-PnP, DPIR and IRCNN) on the full CBSD68 dataset in Table 5.

Original: For fair comparison with RED, we also expose Table 6 the PSNR calculated on the Y channel only.
Revised: For fair comparison with RED, we also expose Table 6 the PSNR calculated on the Y channel only.

Original: An additional visual comparison is ﬁnally shown in Figure 5.
Revised: An additional visual comparison is ﬁnally shown in Figure 6.

Original: Details and comments are given in the corresponding captions.
Revised: Details and comments are given in the corresponding captions.

Original: [Table] [Table]
Revised: [Table] [Table]

Original: [Table][Table] [Table] [Table]
Revised: [Table]

Original: [Table][Table] [Table]
Revised: [Table]

Original: [Table] [Table]
Revised: [Table] [Table]

Original: our method in this set of conditions.
Revised: our method in this set of conditions.

Original: However, over the variety of kernels and noise levels, and in particular for motion blur, our method generally outperforms RED.
Revised: However, over the variety of kernels and noise levels, and in particular for motion blur, our method generally outperforms RED.

Original: [Figure]
Revised: [Figure]

Original: G. S UPER - RESOLUTION
Revised: G. S UPER - RESOLUTION

Original: We also present additional super-resolution experiments.
Revised: We also present additional super-resolution experiments.

Original: We realize a full PSNR performance comparison on the Set3c dataset Table 7.
Revised: We realize a full PSNR performance comparison on the Set3c dataset Table 7.

Original: We show additional visual comparisons between methods
Revised: We show additional visual comparisons between methods

Original: Figure 6 and Figure 7.
Revised: Figure 7 and Figure 8.

Original: Details and comments are given in the corresponding captions.
Revised: Details and comments are given in the corresponding captions.

Original: [Table]
Revised: [Table]

Original: all methods at levels
Revised: all methods at levels

Revised: [Table]

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: G. I NPAINTING ( WITH NON - DIFFERENTIABLE DATA  TERM )
Revised: G. I NPAINTING ( WITH NON - DIFFERENTIABLE DATA - FIDELITY TERM )

Original: We now propose to apply our PnP scheme to image inpainting with the degradation model
Revised: We now propose to apply our PnP scheme to image inpainting with the degradation model

Original: [Equation]
Revised: [Equation]

Original: where A is a diagonal matrix with values in { 0 , 1 } .
Revised: where A is a diagonal matrix with values in { 0 , 1 } .

Original: For inpainting, no noise is added to the degraded image.
Revised: For inpainting, no noise is added to the degraded image.

Original: In this context, the data term is the indicator function of A − 1 ( { y } )
Revised: In this context, the data-ﬁdelity term is the indicator function of A − 1 ( { y } )

Original: = { x | Ax = y } : f ( x ) =
Revised: = { x | Ax = y } : f ( x ) =

Original: ı A − 1 ( { y } ) (which, by deﬁnition, equals + ∞ on A − 1 ( { y } ) and 0 elsewhere).
Revised: ı A − 1 ( { y } ) (which, by deﬁnition, equals + ∞ on A − 1 ( { y } ) and 0 elsewhere).

Original: Despite being non differentiable, f still veriﬁes the assumptions of Theorems 1 and 2 and convergence is theoretically ensured.
Revised: Despite being non differentiable, f still veriﬁes the assumptions of Theorems 1 and 2 and convergence is theoretically ensured.

Original: The proximal map becomes the orthogonal projection
Revised: The proximal map becomes the orthogonal projection

Original: Π A − 1 ( { y } )
Revised: Π A − 1 ( { y } )

Original: [Equation]
Revised: [Equation]

Original: In our experiments, the diagonal of A is ﬁlled with Bernoulli random variables with parameter p = 0 .
Revised: In our experiments, the diagonal of A is ﬁlled with Bernoulli random variables with parameter p = 0 .

Original: We run our PnP algorithm with σ = 10 / 255 .
Revised: We run our PnP algorithm with σ = 10 / 255 .

Original: Given the form of f , we do not use the backtracking strategy and keep a ﬁxed stepsize.
Revised: Given the form of f , we do not use the backtracking strategy and keep a ﬁxed stepsize.

Original: Even if we do not exactly know the Lipschitz constant of ∇ g σ , we observed in Figure 4 that, for small noise, it was almost always estimated as slightly larger than 1 .
Revised: Even if we do not exactly know the Lipschitz constant of ∇ g σ , we observed in Figure 4 that, for small noise, it was almost always estimated as slightly larger than 1 .

Original: We thus choose λτ = 1 and empirically conﬁrm convergence with this choice in follow-up experiments (see Figure 8).
Revised: We thus choose λτ = 1 and empirically conﬁrm convergence with this choice in follow-up experiments (see Figure 9).

Original: The algorithm is initialized with x 0 = y + 0 .
Revised: The algorithm is initialized with x 0 = y + 0 .

Original: 5 ∗ (Id − A )
Revised: 5 ∗ (Id − A )

Original: y (masked pixels with value 0 . 5 ) and terminates when the number of iterations exceeds K = 100 .
Revised: y (masked pixels with value 0 . 5 ) and terminates when the number of iterations exceeds K = 100 .

Original: We found it useful to run the ﬁrst 10 iterations of the algorithm at larger noise level σ = 50 / 255 .
Revised: We found it useful to run the ﬁrst 10 iterations of the algorithm at larger noise level σ = 50 / 255 .

Original: As y does not have noise, we found preferable not to run the last extra gradient pass from Algorithm 1.
Revised: As y does not have noise, we found preferable not to run the last extra gradient pass from Algorithm 1.

Original: We show inpainting results on set3C images Figure 8.
Revised: We show inpainting results on set3C images Figure 9.

Original: Our PnP restores the input images with high accuracy, including its small details.
Revised: Our PnP restores the input images with high accuracy, including its small details.

Original: Furthermore, convergence of the residual at rate O ( 1 k ) is empirically conﬁrmed.
Revised: Furthermore, convergence of the residual at rate O ( 1 k ) is empirically conﬁrmed.

Original: [Figure]
Revised: [Figure]

Original: In  of γ
Revised: In this section we study more deeply the inﬂuence of the parameters involved in the GS-PnP algorithm. Figure 10 and Figure 11 respectively plot the average PSNR when deblurring the CBSD images with different values λ ν and σ/ν , and ﬁxed ν = 0 . Both parameters control the strength of the regularization.

Original: [Equation]
Revised: We observe that λ ν and σ have a similar inﬂuence on the output: for small λ ν or small σ , the regularization involved by the denoising pass is not sufﬁcient to counteract the noise ampliﬁcation done by the proximal steps with large τ (recall that when τ → ∞ , Prox τf tends to the pseudo-inverse of A ). On the contrary, as expected, increasing λ and σ tends to over-smooth the output result. On a single image, we also vary the main parameters of both GS-PnP and RED (Figure 12). For fair comparison, the PSNR is computed on the luminance channel only.

Revised: This experiment conﬁrms that, when manually optimizing the parameters for both methods, the PSNR results obtained with

Revised: GS-PnP and RED remain close, as already observed in Table 6.

Original: [Figure] 
Revised: [Figure] [Figure] [Figure]

Original: [Equation]
Revised: In Figure we examines the robustness of the method to the initialization. As can be seen on this experiment, the output image does not change much even for relatively large perturbation of the initialization. We thus observe a robustness to the initialization, both in terms of visual aspect and

Original: [Figure]
Revised: PSNR. We also observed that initializing with a uniform image does not change the output of the algorithm. We suggest that the robustness to the initialization comes from the ﬁrst proximal steps on the data-ﬁdelity term (with a large τ ), which prevents the algorithm to be stuck in a poor local minimum. Note that it is the backtracking procedure that allows to use large τ in the beginning of the algorithm.

Original: [Equation]
Revised: [Equation]

Revised: H DRUN ET light ARCHITECTURE