SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Samuel Hurault
Univ. Bordeaux, CNRS, INRIA, Bordeaux INP, IMB, UMR 5251
samuel.hurault@math.u-bordeaux.fr
Ulugbek Kamilov
kamilov@wustl.edu
Arthur Leclaire
LTCI, Telecom Paris, IP Paris
arthur.leclaire@telecom-paris.fr
Nicolas Papadakis
nicolas.papadakis@math.u-bordeaux.fr
Plug-and-Play (PnP) methods are efficient iterative algorithms for solving ill-posed image inverse problems. PnP methods are obtained by using deep Gaussian denoisers instead of the proximal operator or the gradient-descent step within proximal algorithms. Current PnP schemes rely on data-fidelity terms that have either Lipschitz gradients or closed-form proximal operators, which is not applicable to Poisson inverse problems. Based on the observation that the Gaussian noise is not the adequate noise model in this setting, we propose to generalize PnP using the Bregman Proximal Gradient (BPG) method. BPG replaces the Euclidean distance with a Bregman divergence that can better capture the smoothness properties of the problem. We introduce the Bregman Score Denoiser specifically parametrized and trained for the new Bregman geometry and prove that it corresponds to the proximal operator of a nonconvex potential. We propose two PnP algorithms based on the Bregman Score Denoiser for solving Poisson inverse problems. Extending the convergence results of BPG in the nonconvex settings, we show that the proposed methods converge, targeting stationary points of an explicit global functional. Experimental evaluations conducted on various Poisson inverse problems validate the convergence results and showcase effective restoration performance.
Ill-posed image inverse problems are classically formulated with a minimization problem of the form
$$
\underset {x \in \mathbb {R} ^ {n}} {\arg \min } \lambda f (x) + g (x) \tag {1}
$$
where  $f$  is a data-fidelity term,  $g$  a regularization term, and  $\lambda > 0$  a regularization parameter. The data-fidelity term is generally written as the negative log-likelihood  $f(x) = -\log p(y|x)$  of the
probabilistic observation model chosen to describe the physics of an acquisition  $y \in \mathbb{R}^m$  from linear measurements  $Ax$  of an image  $x \in \mathbb{R}^n$ . In applications such as Positron Emission Tomography (PET) or astronomical CCD cameras [Bertero et al., 2009], where images are obtained by counting particles (photons or electrons), it is common to use the Poisson noise model  $y \sim \mathcal{P}(\alpha Ax)$  with parameter  $\alpha > 0$ . The corresponding negative log-likelihood corresponds to the Kullback-Leibler divergence
$$
f (x) = \sum_ {i = 1} ^ {m} y _ {i} \log \left(\frac {y _ {i}}{\alpha (A x) _ {i}}\right) + \alpha (A x) _ {i} - y _ {i}. \tag {2}
$$
The minimization of (1) can be addressed with proximal splitting algorithms [Combettes and Pesquet, 2011]. Depending on the properties of the functions  $f$  and  $g$ , they consist in alternatively evaluating the proximal operator and/or performing a gradient-descent step on  $f$  and  $g$ .
Plug-and-Play (PnP) [Venkatakrishnan et al., 2013] and Regularization-by-Denoising (RED) [Romano et al., 2017] methods build on proximal splitting algorithms by replacing the proximal or gradient descent updates with off-the-shelf Gaussian denoising operators, typically deep neural networks trained to remove Gaussian noise. The intuition behind PnP/RED is that the proximal (resp. gradient-descent) mapping of  $g$  writes as the maximum-a-posteriori (MAP) (resp. posterior mean) estimation from a Gaussian noise observation, under prior  $p = \exp(-g)$ . A remarkable property of these priors is that they are decoupled from the degradation model represented by  $f$  in the sense that one learned prior  $p$  can serve as a regularizer for many inverse problems. When using deep denoisers corresponding to exact gradient-step [Hurault et al., 2021, Cohen et al., 2021] or proximal [Hurault et al., 2022] maps, RED and PnP methods become real optimization algorithms with known convergence guarantees.
However, for Poisson noise the data-fidelity term (2) is neither smooth with a Lipschitz gradient, nor proximable for  $A \neq \mathrm{Id}$  (meaning that its proximal operator cannot be computed in a closed form), which limits the use of standard splitting procedures. Bauschke et al. [2017] addressed this issue by introducing a Proximal Gradient Descent (PGD) algorithm in the Bregman divergence paradigm, called Bregman Proximal Gradient (BPG). The benefit of BPG is that the smoothness condition on  $f$  for sufficient decrease of PGD is replaced by the "NoLip" condition "Lh - f is convex" for a convex potential  $h$ . For instance, Bauschke et al. [2017] show that the data-fidelity term (2) satisfies the NoLip condition for the Burg's entropy  $h(x) = -\sum_{i=1}^{n} \log(x_i)$ .
Our primary goal is to exploit the BPG algorithm for minimizing (1) using PnP and RED priors. This requires an interpretation of the plug-in denoiser as a Bregman proximal operator.
$$
\operatorname {P r o x} _ {g} ^ {h} (y) = \underset {x} {\arg \min } g (x) + D _ {h} (x, y), \tag {3}
$$
where  $D_h(x,y)$  is the Bregman divergence associated with  $h$ . In Section 3, we show that by selecting a suitable noise model, other than the traditional Gaussian one, the MAP denoiser can be expressed as a Bregman proximal operator. Remarkably, the corresponding noise distribution belongs to an exponential family, which allows for a closed-form posterior mean (MMSE) denoiser generalizing the Tweedie's formula. Using this interpretation, we derive a RED prior tailored to the Bregman geometry. By presenting a prior compatible with the noise model, we highlight the limitation of the decoupling between prior and data-fidelity suggested in the existing PnP literature.
In order to safely use our MAP and MMSE denoisers in the BPG method, we introduce the Bregman Score denoiser, which generalizes the Gradient Step denoiser from [Hurault et al., 2021, Cohen et al., 2021]. Our denoiser provides an approximation of the log prior of the noisy distribution of images. Moreover, based on the characterization of the Bregman Proximal operator from Gribonval and Nikolova [2020], we show under mild conditions that our denoiser can be expressed as the Bregman proximal operator of an explicit nonconvex potential.
In Section 4, we use the Bregman Score Denoiser within RED and PnP methods and propose the B-RED and B-PnP algorithms. Elaborating on the results from Bolte et al. [2018] for the BPG algorithm in the nonconvex setting, we demonstrate that RED-BPG and PnP-BPG are guaranteed to converge towards stationary points of an explicit functional. We finally show in Section 5 the relevance of the proposed framework in the context of Poisson inverse problems.
Poisson Inverse Problems A variety of methods have been proposed to solve Poisson image restoration from the Bayesian variational formulation (1) with Poisson data-fidelity term (2). Although
$f$  is convex, this is a challenging optimization problem as  $f$  is non-smooth and does not have a closed-form proximal operator for  $A \neq \mathrm{Id}$ , thus precluding the direct application of splitting algorithms such as Proximal Gradient Descent or ADMM. Moreover, we wish to regularize (1) with an explicit denoising prior, which is generally nonconvex. PIDAL [Figueiredo and Bioucas-Dias, 2009, 2010] and related methods [Ng et al., 2010, Setzer et al., 2010] solve (2) using modified versions of the alternating direction method of multipliers (ADMM). As  $f$  is proximable when  $A = \mathrm{Id}$ , the idea is to add a supplementary constraint in the minimization problem and to adopt an augmented Lagrangian framework. Figueiredo and Bioucas-Dias [2010] prove the convergence of their algorithm with convex regularization. However, no convergence is established for nonconvex regularization. Boulanger et al. [2018] adopt the primal-dual PDHG algorithm which also splits  $A$  from the  $f$  update. Once again, there is not convergence guarantee for the primal-dual algorithm with nonconvex regularization.
Plug-and-Play (PnP) PnP methods were successfully used for solving a variety of IR tasks by including deep denoisers into different optimization algorithms, including Gradient Descent [Romano et al., 2017], Half-Quadratic-Splitting [Zhang et al., 2017, 2021], ADMM [Ryu et al., 2019, Sun et al., 2021], and PGD [Kamilov et al., 2017, Terris et al., 2020]. Variations of PnP have been proposed to solve Poisson inverse problems. Rond et al. [2016] use PnP-ADMM and approximates, at each iteration, the non-tractable proximal operator of the data-fidelity term (2) with an inner minimization procedure. Sanghvi et al. [2022] propose a PnP version of the PIDAL algorithm which is also unrolled for image deconvolution. Theoretical convergence of PnP algorithms with deep denoisers has recently been addressed by a variety of studies [Ryu et al., 2019, Sun et al., 2021, Terris et al., 2020] (see also a review in Kamilov et al. [2023]). Most of these works require non-realistic or sub-optimal constraints on the deep denoiser, such as nonexpansiveness. More recently, convergence was addressed by making PnP genuine optimization algorithms again. This is done by building deep denoisers as exact gradient descent operators (Gradient-Step denoiser) [Cohen et al., 2021, Hurault et al., 2021] or exact proximal operators [Hurault et al., 2022, 2023]. These PnP algorithms thus minimize an explicit functional (2) with an explicit (nonconvex) deep regularization.
Bregman optimization Bauschke et al. [2017] replace the smoothness condition of PGD by the NoLip assumption (4) with a Bregman generalization of PGD called Bregman Proximal Gradient (BPG). In the nonconvex setting, Bolte et al. [2018] prove global convergence of the algorithm to a critical point of (1). The analysis however requires assumptions that are not verified by the Poisson data-fidelity term and Burg's entropy Bregman potential. Al-Shabili et al. [2022] considered unrolled Bregman PnP and RED algorithms, but without any theoretical convergence analysis. Moreover, the interaction between the data fidelity, the Bregman potential, and the denoiser was not explored.
3 Bregman denoising prior
The overall objective of this work is to efficiently solve ill-posed image restoration (IR) problems involving a data-fidelity term  $f$  verifying the NoLip assumption for some convex potential  $h$
$$
\text {N o L i p} \quad \text {T h e r e i s} L > 0 \text {s u c h t h a t} L h - f \text {i s c o n v e x o n i n t d o m} h. \tag {4}
$$
PnP provides an elegant framework for solving ill-posed inverse problems with a denoising prior. However, the intuition and efficiency of PnP methods inherit from the fact that Gaussian noise is well suited for the Euclidean  $L_{2}$  distance, the latter naturally arising in the MAP formulation of the Gaussian denoising problem. When the Euclidean distance is replaced by a more general Bregman divergence, the noise model needs to be adapted accordingly for the prior.
In Section 3.1, we first discuss the choice of the noise model associated to a Bregman divergence, leading to Bregman formulations of the MAP and MMSE estimators. Then we introduce in Section 3.2 the Bregman Score Denoiser that will be used to regularize the inverse problem (1).
Notations For convenience, we assume throughout our analysis that the convex potential  $h: C_h \subseteq \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$  is  $\mathcal{C}^2$  and of Legendre type (definition in Appendix A.1). Its convex conjugate  $h^*$  is then also  $\mathcal{C}^2$  of Legendre type.  $D_h(x, y)$  denotes its associated Bregman divergence
$$
D _ {h}: \mathbb {R} ^ {n} \times i n t \operatorname {d o m} h \rightarrow [ 0, + \infty ]: (x, y) \rightarrow \left\{\begin{array}{l l}h (x) - h (y) - \langle \nabla h (y), x - y \rangle&\text {i f} x \in \operatorname {d o m} (h)\\+ \infty&\text {o t h e r w i s e .}\end{array}\right. \tag {5}
$$
3.1 Bregman noise model
We consider the following observation noise model, referred to as Bregman noise<sup>1</sup>,
$$
\text {f o r} x, y \in \operatorname {d o m} (h) \times \operatorname {i n t} \operatorname {d o m} (h) \quad p (y | x) := \exp (- \gamma D _ {h} (x, y) + \rho (x)). \tag {6}
$$
We assume that there is  $\gamma > 0$  and a normalizing function  $\rho: \mathrm{dom}(h) \to \mathbb{R}$  such that the expression (6) defines a probability measure. For instance, for  $h(x) = \frac{1}{2} ||x||^2$ ,  $\gamma = \frac{1}{\sigma^2}$  and  $\rho = 0$ , we retrieve the Gaussian noise model with variance  $\sigma^2$ . As shown in Section 5, for  $h$  given by Burg's entropy,  $p(y|x)$  corresponds to a multivariate Inverse Gamma ( $\mathcal{IG}$ ) distribution.
Given a noisy observation  $y \in \text{int} \, \text{dom}(h)$ , i.e. a realization of a random variable  $Y$  with conditional probability  $p(y|x)$ , we now consider two optimal estimators of  $x$ , the MAP and the posterior mean.
Maximum-A-Posteriori (MAP) estimator The MAP denoiser selects the mode of the a-posteriori probability distribution  $p(x|y)$ . Given the prior  $p_X$ , it writes
$$
\hat {x} _ {M A P} (y) = \underset {x} {\arg \min } - \log p (x | y) = \underset {x} {\arg \min } - \log p _ {X} (x) - \log p (y | x) = \operatorname {P r o x} _ {- \frac {1}{\gamma} (\rho + \log p _ {X})} ^ {h} (y). \tag {7}
$$
Under the Bregman noise model (6), the MAP denoiser writes as the Bregman proximal operator (see relation (3)) of  $-\frac{1}{\gamma} (\log p_X + \rho)$ . This acknowledges for the fact that the introduced Bregman noise is the adequate noise model for generalizing PnP methods within the Bregman framework.
Posterior mean (MMSE) estimator The MMSE denoiser is the expected value of the posterior probability distribution and the optimal Bayes estimator for the  $L_{2}$  score. Note that our Bregman noise conditional probability (6) belongs to the regular exponential family of distributions
$$
p (y | x) = p _ {0} (y) \exp (\langle x, T (y) \rangle - \psi (x)) \tag {8}
$$
with  $T(y) = \gamma \nabla h(y)$ ,  $\psi(x) = \gamma h(x) - \rho(x)$  and  $p_0(y) = \exp(\gamma h(y) - \gamma \langle \nabla h(y), y \rangle)$ . It is shown in [Efron, 2011] (for  $T = \mathrm{Id}$  and generalized in [Kim and Ye, 2021] for  $T \neq \mathrm{Id}$ ) that the corresponding posterior mean estimator verifies a generalized Tweedie formula  $\nabla T(y).\hat{x}_{MMSE}(y) = -\nabla \log p_0(y) + \nabla \log p_Y(y)$ , which translates to (see Appendix B for details)
$$
\hat {x} _ {M M S E} (y) = \mathbb {E} [ x | y ] = y - \frac {1}{\gamma} \left(\nabla^ {2} h (y)\right) ^ {- 1} \cdot \nabla \left(- \log p _ {Y}\right) (y). \tag {9}
$$
Note that for the Gaussian noise model, we have  $h(x) = \frac{1}{2} ||x||^2$ ,  $\gamma = 1 / \sigma^2$  and (9) falls back to the more classical Tweedie formula of the Gaussian posterior mean denoiser  $\hat{x} = y - \sigma^2\nabla (-\log p_Y)(y)$ .
Therefore, given an off-the-shelf "Bregman denoiser"  $\mathcal{B}_{\gamma}$  specially devised to remove Bregman noise (6) of level  $\gamma$ , if the denoiser approximates the posterior mean  $\mathcal{B}_{\gamma}(y)\approx \hat{x}_{MMSE}(y)$ , then it provides an approximation of the score  $-\nabla \log p_{Y}(y)\approx \gamma \nabla^{2}h(y).(y - \mathcal{B}_{\gamma}(y))$
3.2 Bregman Score Denoiser
Based on previous observations, we propose to define a denoiser following the form of the MMSE (9)
$$
\mathcal {B} _ {\gamma} (y) = y - \left(\nabla^ {2} h (y)\right) ^ {- 1}. \nabla g _ {\gamma} (y), \tag {10}
$$
with  $g_{\gamma}:\mathbb{R}^n\to \mathbb{R}$  a nonconvex potential parametrized by a neural network. Such a denoiser corresponds to the Bregman generalization of the Gaussian Noise Gradient-Step denoiser proposed in [Hurault et al., 2021, Cohen et al., 2021].
When  $\mathcal{B}_{\gamma}$  is trained as a denoiser for the associated Bregman noise (6) with  $L_{2}$  loss, it approximates the optimal estimator for the  $L_{2}$  score, precisely the MMSE (9). Comparing (10) and (9), we get  $\nabla g_{\gamma} \approx -\frac{1}{\gamma} \nabla \log p_{Y}$ , i.e. the score is properly approximated with an explicit conservative vector field. We refer to this denoiser as the Bregman Score denoiser.
Is the Bregman Score Denoiser a Bregman proximal operator? We showed in relation (7) that the optimal Bregman MAP denoiser  $\hat{x}_{MAP}$  is a Bregman proximal operator. We want to generalize this
<sup>1</sup>The Bregman divergence being non-symmetric, the order of the variables  $(x,y)$  in  $D_h$  is important. Distributions of the form (6) with reverse order in  $D_h$  have been characterized in [Banerjee et al., 2005] but this analysis does not apply here.
property to our Bregman denoiser (10). When trained with  $L_{2}$  loss, the denoiser should approximate the MMSE rather than the MAP. For Gaussian noise, Gribonval [2011] re-conciliates the two views by showing that the Gaussian MMSE denoiser actually writes as an Euclidean proximal operator.
Similarly, extending the characterization from [Gribonval and Nikolova, 2020] of Bregman proximal operators, we now prove that, under some convexity conditions, the proposed Bregman Score Denoiser (10) explicitly writes as the Bregman proximal operator of a nonconvex potential.
Proposition 1 (Proof in Appendix C). Let  $h$  be  $\mathcal{C}^2$  and of Legendre type. Let  $g_{\gamma}: \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$  proper and differentiable and  $\mathcal{B}_{\gamma}(y): \operatorname{int}\operatorname{dom}(h) \to \mathbb{R}^n$  defined in (10). Assume  $\operatorname{Im}(\mathcal{B}_{\gamma}) \subset \operatorname{int}\operatorname{dom}(h)$ . With  $\psi_{\gamma}: \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$  defined by
$$
\psi_ {\gamma} (y) = \left\{ \begin{array}{l l} - h (y) + \langle \nabla h (y), y \rangle - g _ {\gamma} (y) & \text {i f} y \in \operatorname {i n t} \operatorname {d o m} (h) \\ + \infty & \text {o t h e r w i s e .} \end{array} \right. \tag {11}
$$
suppose that  $\psi_{\gamma} \circ \nabla h^{*}$  is convex on int  $\mathrm{dom}(h^{*})$ . Then for  $\phi_{\gamma}: \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$  defined by
$$
\phi_ {\gamma} (x) := \left\{ \begin{array}{l l} g _ {\gamma} (y) - D _ {h} (x, y) f o r y \in \mathcal {B} _ {\gamma} ^ {- 1} (x) & i f x \in \operatorname {I m} (\mathcal {B} _ {\gamma}) \\ + \infty & o t h e r w i s e \end{array} \right. \tag {12}
$$
we have that for each  $y \in \operatorname{int} \operatorname{dom}(h)$
$$
\mathcal {B} _ {\gamma} (y) \in \underset {x \in \mathbb {R} ^ {n}} {\arg \min } \left\{D _ {h} (x, y) + \phi_ {\gamma} (x) \right\} \tag {13}
$$
Remark 1. Note that the order in the Bregman divergence is important in order to fit the definition of the Bregman proximal operator (3). In this order, [Gribonval and Nikolova, 2020, Theorem 3] does not directly apply. We propose instead in Appendix C a new version of their main theorem.
Remark 2. As shown in Appendix C (equation (64)),  $\mathcal{B}_{\gamma}(y)$  can be written as  $\mathcal{B}_{\gamma}(y) = \nabla (\psi_{\gamma} \circ \nabla h^{*}) \circ \nabla h(y)$  so that the Jacobian  $J_{\mathcal{B}_{\gamma}}(y) = \nabla^2 h(y). \nabla^2 (\psi_{\gamma} \circ \nabla h^{*})(\nabla h(y))$ . The hypothesis of convexity of  $\psi_{\gamma} \circ \nabla h^{*}$  on  $\operatorname{int} \operatorname{dom}(h^{*})$  is thus equivalent to the fact that the Jacobian of  $\mathcal{B}_{\gamma}$  is positive semi-definite on  $\operatorname{int} \operatorname{dom}(h)$ .
This proposition generalizes the result of [Hurault et al., 2022, Prop. 1] to any Bregman geometry, which proves that the Gradient-Step Gaussian denoiser writes as an Euclidean proximal operator when  $\psi_{\gamma} \circ \nabla h^{*}(x) = \frac{1}{2} ||x||^{2} - g_{\gamma}(x)$  is convex. More generally, as exhibited for Poisson inverse problems in Section 5, such a convexity condition translates to a constraint on the deep potential  $g_{\gamma}$ .
To conclude, the Bregman Score Denoiser provides, via exact gradient or proximal mapping, two distinct explicit nonconvex priors  $g_{\gamma}$  and  $\phi_{\gamma}$  that can be used for subsequent PnP image restoration.
4 Plug-and-Play (PnP) image restoration with Bregman Score Denoiser
We now regularize inverse problems with the explicit prior provided by the Bregman Score Denoiser (10). Properties of the Bregman Proximal Gradient (BPG) algorithm are recalled in Section 4.1. We show the convergence of our two B-RED and B-PnP algorithms in Sections 4.2 and 4.3.
4.1 Bregman Proximal Gradient (BPG) algorithm
Let  $F$  and  $\mathcal{R}$  be two proper and lower semi-continuous functions with  $F$  of class  $\mathcal{C}^1$  on int  $\mathrm{dom}(h)$ . Bauschke et al. [2017] propose to minimize  $\Psi = F + \mathcal{R}$  using the following BPG algorithm:
$$
x ^ {k + 1} \in \underset {x \in \mathbb {R} ^ {n}} {\arg \min } \left\{\mathcal {R} (x) + \langle x - x ^ {k}, \nabla F \left(x ^ {k}\right) \rangle + \frac {1}{\tau} D _ {h} \left(x, x ^ {k}\right) \right\}. \tag {14}
$$
Recalling the general expression of proximal operators defined in relation (3), when  $\nabla h(x_k) - \tau \nabla F(x_k) \in \mathrm{dom}(h^*)$ , the previous iteration can be written as (see Appendix D)
$$
x ^ {k + 1} \in \operatorname {P r o x} _ {\tau \mathcal {R}} ^ {h} \circ \nabla h ^ {*} (\nabla h - \tau \nabla F) (x _ {k}). \tag {15}
$$
With formulation (15), the BPG algorithm generalizes the Proximal Gradient Descent (PGD) algorithm in a different geometry defined by  $h$ .
Convergence of BPG If  $F$  verifies the NoLip condition (4) for some  $L_{F} > 0$  and if  $\tau < \frac{1}{L_F}$ , one can prove that the objective function  $\Psi$  decreases along the iterates (15). Global convergence of the
iterates for nonconvex  $F$  and  $\mathcal{R}$  is also shown in [Bolte et al., 2018]. However, Bolte et al. [2018] take assumptions on  $F$  and  $h$  that are not satisfied in the context of Poisson inverse problems. For instance,  $h$  is assumed strongly convex on the full domain  $\mathbb{R}^n$  which is not satisfied by Burg's entropy. Additionally,  $F$  is assumed to have Lipschitz-gradient on bounded subset of  $\mathbb{R}^n$ , which is not true for  $F$  the Poisson data-fidelity term (2). Following the same structure of their proof, we extend in Appendix D.1 (Proposition 3 and Theorem 3) the convergence theory from [Bolte et al., 2018] with the more general Assumption 1 below, which is verified for Poisson inverse problems (Appendix E.4).
Application to the IR problem In what follows, we consider two variants of BPG for minimizing (1) with gradient updates on the data-fidelity term  $f$ . These algorithms respectively correspond to Bregman generalizations of the RED Gradient-Descent (RED-GD) [Romano et al., 2017] and the Plug-and-Play PGD algorithms. For the rest of this section, we consider the following assumptions
(i)  $h:C_h\to \mathbb{R}\cup \{+\infty \}$  is of class  $\mathcal{C}^2$  and of Legendre-type.
(ii)  $f:\mathbb{R}^n\to \mathbb{R}\cup \{+\infty \}$  is proper, lower-bounded, coercive, of class  $\mathcal{C}^1$  on int  $\mathrm{dom}(h)$ , with  $\operatorname {dom}(h)\subset \operatorname {dom}(f)$ , and is subanalytic.
(iii)  $\mathbf{NoLip}: L_f h - f$  is convex on  $\operatorname{int} \operatorname{dom}(h)$ .
(iv)  $h$  is assumed strongly convex on any bounded convex subset of its domain and for all  $\alpha > 0$ ,  $\nabla h$  and  $\nabla f$  are Lipschitz continuous on  $\{x \in \operatorname{dom}(h), \Psi(x) \leq \alpha\}$ .
(v)  $g_{\gamma}$  given by the Bregman Score Denoiser (10) and its associated  $\phi_{\gamma}$  obtained from Proposition 1 are lower-bounded and subanalytic.