SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Even though the Poisson data-fidelity term (2) is convex, our convergence results also hold for more general nonconvex data-fidelity terms. Assumption (iv) generalizes [Bolte et al., 2018, Assumption D] and allows to prove global convergence of the iterates. The subanalytic assumption, defined in Appendix A.3, is a sufficient condition for the Kurdyka-Lojasiewicz (KL) property to be verified [Bolte et al., 2007]. The latter, also defined in Appendix A.3, can be interpreted as the fact that, up to a reparameterization, the function is sharp. The KL property is widely used in nonconvex optimization [Attouch et al., 2013, Ochs et al., 2014, Bolte et al., 2018]. As detailed in Appendix A.3, the subanalytic assumption of  $f$ ,  $g_{\gamma}$  and  $\phi_{\gamma}$  allows for the Kurdyka-Lojasiewicz (KL) property to be verified by the objective functions  $f + g_{\gamma}$  and  $f + \phi_{\gamma}$ .
4.2 Bregman Regularization-by-Denoising (B-RED)
We first generalize the RED Gradient-Descent (RED-GD) algorithm [Romano et al., 2017] in the Bregman framework. Classically, RED-GD is a simple gradient-descent algorithm applied to the functional  $\lambda f + g_{\gamma}$  where the gradient  $\nabla g_{\gamma}$  is assumed to be implicitly given by an image denoiser  $\mathcal{B}_{\gamma}$  (parametrized by  $\gamma$ ) via  $\nabla g_{\gamma} = \operatorname{Id} - \mathcal{B}_{\gamma}$ . Instead, our Bregman Score Denoiser (10) provides an explicit regularizing potential  $g_{\gamma}$  whose gradient approximates the score via the Tweedie formula (9). We propose to minimize  $F_{\lambda,\gamma} = \lambda f + g_{\gamma}$  on  $\operatorname{dom}(h)$  using the Bregman Gradient Descent algorithm
$$
x _ {k + 1} = \nabla h ^ {*} (\nabla h - \tau \nabla F _ {\lambda , \gamma}) (x _ {k}) \tag {16}
$$
which writes in a more general version as the BPG algorithm (14) with  $\mathcal{R} = 0$
$$
x _ {k + 1} = \underset {x \in \mathbb {R} ^ {n}} {\arg \min } \left\{\langle x - x _ {k}, \lambda \nabla f (x _ {k}) + \nabla g _ {\gamma} (x _ {k}) \rangle + \frac {1}{\tau} D _ {h} (x, x _ {k}) \right\}. \tag {17}
$$
As detailed in Appendix E.4, in the context of Poisson inverse problems, for  $h$  being Burg's entropy,  $F_{\lambda,\gamma} = \lambda f + g_{\gamma}$  verifies the NoLip condition only on bounded convex subsets of  $\operatorname{dom}(h)$ . Thus we select  $C$  a non-empty closed bounded convex subset of  $\overline{\operatorname{dom}(h)}$ . For the algorithm (17) to be well-posed and to verify a sufficient decrease of  $(F_{\lambda,\gamma}(x^k))$ , the iterates need to verify  $x_k \in C$ . We propose to modify (17) as the Bregman version of Projected Gradient Descent, which corresponds to the BPG algorithm (14) with  $\mathcal{R} = i_C$ , the characteristic function of the set  $C$ :
$$
\left(\mathbf {B} - \mathbf {R E D}\right) \quad x ^ {k + 1} \in T _ {\tau} \left(x _ {k}\right) = \underset {x \in \mathbb {R} ^ {n}} {\arg \min } \left\{i _ {C} (x) + \left\langle x - x ^ {k}, \nabla F _ {\lambda , \gamma} \left(x ^ {k}\right) \right\rangle + \frac {1}{\tau} D _ {h} \left(x, x ^ {k}\right) \right\}. \tag {18}
$$
For general convergence of B-RED, we need the following assumptions
(i)  $\mathcal{R} = i_C$ , with  $C$  a non-empty closed, bounded, convex and semi-algebraic subset of  $\overline{\operatorname{dom}(h)}$  such that  $C \cap \operatorname{int}\operatorname{dom}(h) \neq \emptyset$ .
(ii)  $g_{\gamma}$  has Lipschitz continuous gradient and there is  $L_{\gamma} > 0$  such that  $L_{\gamma}h - g_{\gamma}$  is convex on  $C \cap \operatorname{int}\operatorname{dom}(h)$ .
In [Bolte et al., 2018, Bauschke et al., 2017], the NoLip constant  $L$  needs to be known to set the stepsize of the BPG algorithm as  $\tau L < 1$ . In practice, the NoLip constant which depends on  $f$ ,  $g_{\gamma}$  and  $C$  is either unknown or over-estimated. In order to avoid small stepsize, we adapt the backtracking strategy of [Beck, 2017, Chapter 10] to automatically adjust the stepsize while keeping convergence guarantees. Given  $\gamma \in (0,1)$ ,  $\eta \in [0,1)$  and an initial stepsize  $\tau_0 > 0$ , the following backtracking update rule on  $\tau$  is applied at each iteration  $k$ :
$$
\text {w h i l e} F _ {\lambda , \gamma} \left(x _ {k}\right) - F _ {\lambda , \gamma} \left(T _ {\tau} \left(x _ {k}\right)\right) <   \frac {\gamma}{\tau} D _ {h} \left(T _ {\tau} \left(x _ {k}\right), x _ {k}\right), \quad \tau \longleftarrow \eta \tau . \tag {19}
$$
Using the general nonconvex convergence analysis of BPG realized in Appendix D.1, we can show sufficient decrease of the objective and convergence of the iterates of B-RED.
Theorem 1 (Proof in Appendix D.2). Under Assumption 1 and Assumption 2, the iterates  $(x_{k})$  given by the B-RED algorithm (18) with the backtracking procedure (19) decrease  $F_{\lambda,\gamma}$  and converge to a critical point of  $\Psi = i_{C} + F_{\lambda,\gamma}$  with rate  $\min_{0 \leq k \leq K} D_{h}(x^{k+1}, x^{k}) = O(1 / K)$ .
4.3 Bregman Plug-and-Play (B-PnP)
We now consider the equivalent of PnP Proximal Gradient Descent algorithm in the Bregman framework. Given a denoiser  $\mathcal{B}_{\gamma}$  with  $\mathrm{Im}(\mathcal{B}_{\gamma}) \subset \mathrm{dom}(h)$  and  $\lambda > 0$  such that  $\mathrm{Im}(\nabla h - \lambda \nabla f) \subseteq \mathrm{dom}(\nabla h^{*})$ , it writes
$$
(\mathbf {B} - \mathbf {P n P}) \quad x ^ {k + 1} = \mathcal {B} _ {\gamma} \circ \nabla h ^ {*} (\nabla h - \lambda \nabla f) (x _ {k}). \tag {20}
$$
We use again as  $\mathcal{B}_{\gamma}$  the Bregman Score Denoiser (10). With  $\psi_{\gamma}$  defined from  $g_{\gamma}$  as in (11) and assuming that  $\psi_{\gamma} \circ \nabla h^{*}$  is convex on  $\operatorname{int} \operatorname{dom}(h^{*})$ , Proposition 1 states that the Bregman Score denoiser  $\mathcal{B}_{\gamma}$  is the Bregman proximal operator of some nonconvex potential  $\phi_{\gamma}$  verifying (12). The algorithm B-PnP (20) then becomes  $x^{k+1} \in \mathrm{Prox}_{\phi_{\gamma}}^{h} \circ \nabla h^{*}(\nabla h - \lambda \nabla f)(x_{k})$ , which writes as a Bregman Proximal Gradient algorithm, with stepsize  $\tau = 1$ ,
$$
x ^ {k + 1} \in \underset {x \in \mathbb {R} ^ {n}} {\arg \min } \left\{\phi_ {\gamma} (x) + \langle x - x ^ {k}, \lambda \nabla f \left(x ^ {k}\right) \rangle + D _ {h} \left(x, x ^ {k}\right) \right\}. \tag {21}
$$
With Proposition 1, we have  $\mathcal{B}_{\gamma}(y) \in \mathrm{Prox}_{\phi_{\gamma}}^{h}(y)$  i.e. a proximal step on  $\phi_{\gamma}$  with stepsize 1. We are thus forced to keep a fixed stepsize  $\tau = 1$  in the BPG algorithm (21) and no backtracking is possible. Using Appendix D.1, we can show that B-PnP converges towards a stationary point of  $\lambda f + \phi_{\gamma}$ .
Theorem 2 (Proof in Appendix D.3). Assume Assumption 1 and  $\psi_{\gamma} \circ \nabla h^{*}$  strictly convex on  $int \, \mathrm{dom}(h^{*})$ . Then for  $\operatorname{Im}(\nabla h - \lambda \nabla f) \subseteq \operatorname{dom}(\nabla h^{*})$ ,  $\operatorname{Im}(\mathcal{B}_{\gamma}) \subseteq \operatorname{dom}(h)$  and  $\lambda L_{f} < 1$  (with  $L_{f}$  specified in Assumption 1), the iterates  $x_{k}$  given by the B-PnP algorithm (20) decrease  $\lambda f + \phi_{\gamma}$  and converge to a critical point of  $\lambda f + \phi_{\gamma}$  with rate  $\min_{0 \leq k \leq K} D_{h}(x^{k+1}, x^{k}) = O(1 / K)$ .
Remark 3. The condition  $\mathrm{Im}(\mathcal{B}_{\gamma}) \subseteq \mathrm{dom}(h)$  and the required convexity of  $\psi_{\gamma} \circ \nabla h^{*}$  come from Proposition 1 while the condition  $\mathrm{Im}(\nabla h - \lambda \nabla f) \subseteq \mathrm{dom}(\nabla h^{*})$  allows the algorithm B-PnP (20) to be well-posed. These assumptions will be discussed with more details in the context of Poisson image restoration in Section 5.
5 Application to Poisson inverse problems
We consider ill-posed inverse problems involving the Poisson data-fidelity term  $f$  introduced in (2). The Euclidean geometry (i.e.  $h(x) = \frac{1}{2} ||x||^2$ ) does not suit for such  $f$ , as it does not have a Lipschitz gradient. In [Bauschke et al., 2017, Lemma 7], it is shown that an adequate Bregman potential  $h$  (in the sense that there exists  $L_{f}$  such that  $L_{f}h - f$  is convex) for (2) is the Burg's entropy
$$
h (x) = - \sum_ {i = 1} ^ {n} \log \left(x _ {i}\right), \tag {22}
$$
for which  $\mathrm{dom}(h) = \mathbb{R}_{++}^n$  and  $L_{f}h - f$  is convex on int  $\mathrm{dom}(h) = \mathbb{R}_{++}^n$  for  $L_{f}\geq ||y||_{1}$ . For further computation, note that the Burg's entropy (22) satisfies  $\nabla h(x) = \nabla h^{*}(x) = -\frac{1}{x}$  and  $\nabla^2 h(x) = \frac{1}{x^2}$ . The Bregman score denoiser associated to the Burg's entropy is presented in Section 5.1. The corresponding Bregman RED and PnP algorithms are applied to Poisson Image deblurring in Section 5.2.
5.1 Bregman Score Denoiser with Burg's entropy
We now specify the study of Section 3 to the case of the Burg's entropy (22). In this case, the Bregman noise model (6) writes (see Appendix E.1 for detailed calculus)
$$
p (y | x) = \exp (\rho (x) + n \gamma) \prod_ {i = 1} ^ {n} \left(\frac {x _ {i}}{y _ {i}}\right) ^ {\gamma} \exp \left(- \gamma \frac {x _ {i}}{y _ {i}}\right). \tag {23}
$$
For  $\gamma > 1$ , this is a product of Inverse Gamma  $(\mathcal{IG}(\alpha, \beta))$  distributions with parameter  $\beta_i = \gamma x_i$  and  $\alpha_i = \gamma - 1$ . This noise model has mean (for  $\gamma > 2$ )  $\frac{\gamma}{\gamma - 2}x$  and variance (for  $\gamma > 3$ )  $\frac{\gamma^2}{(\gamma - 2)^2(\gamma - 3)}x^2$ . In particular, for large  $\gamma$ , the noise becomes centered on  $x$  with signal-dependent variance  $x^2/\gamma$ . Furthermore, using Burg's entropy (22), the optimal posterior mean (9) and the Bregman Score Denoiser (10) respectively write, for  $y \in \mathbb{R}_{++}^n$ ,
$$
\hat {x} _ {M M S E} (y) = y - \frac {1}{\gamma} y ^ {2} \nabla (- \log p _ {Y}) (y) \tag {24}
$$
$$
\mathcal {B} _ {\gamma} (y) = y - y ^ {2} \nabla g _ {\gamma} (y). \tag {25}
$$
Denoising in practice As Hurault et al. [2021], we chose to parametrize the deep potential  $g_{\gamma}$  as
$$
g _ {\gamma} (y) = \frac {1}{2} | | x - N _ {\gamma} (x) | | ^ {2}, \tag {26}
$$
where  $N_{\sigma}$  is the deep convolutional neural network architecture DRUNet [Zhang et al., 2021] that contains Softplus activations and takes the noise level  $\gamma$  as input.  $\nabla g_{\gamma}$  is computed with automatic differentiation. We train  $\mathcal{B}_{\gamma}$  to denoise images corrupted with random Inverse Gamma noise of level  $\gamma$ , sampled from clean images via  $p(y|x) = \prod_{i=1}^{n} \mathcal{I}\mathcal{G}(\alpha_i, \beta_i)(x_i)$ . To sample  $y_i \sim \mathcal{I}\mathcal{G}(\alpha_i, \beta_i)$ , we sample  $z_i \sim \mathcal{G}(\alpha_i, \beta_i)$  and take  $y_i = 1/z_i$ . Denoting as  $p$  the distribution of a database of clean images, training is performed with the  $L^2$  loss
$$
\mathcal {L} (\gamma) = \mathbb {E} _ {x \sim p, y \sim \mathcal {I G} _ {\gamma} (x)} \left[ \| \mathcal {B} _ {\gamma} (y) - x \| ^ {2} \right]. \tag {27}
$$
Denoising performance We evaluate the performance of the proposed Bregman Score DRUNet (B-DRUNet) denoiser (26). It is trained with the loss (27), with  $1 / \gamma$  uniformly sampled in  $(0, 0.1)$ . More details on the architecture and the training can be found in Appendix E.2. We compare the performance of B-DRUNet (26) with the same network DRUNet directly trained to denoise inverse Gamma noise with  $L_{2}$  loss. Qualitative and quantitative results presented in Figure 1 and Table 1 show that the Bregman Score Denoiser (B-DRUNet), although constrained to be written as (25) with a conservative vector field  $\nabla g_{\gamma}$ , performs on par with the unconstrained denoiser (DRUNet).
γ	10	25	50	100	200
DRUNet	28.42	30.91	32.80	34.76	36.79
B-DRUNet	28.38	30.88	32.76	34.74	36.71
Table 1: Average denoising PSNR performance of Inverse Gamma noise denoisers B-DRUNet and DRUNet on  ${256} \times  {256}$  center-cropped images from the CBSD68 dataset,for various noise levels  $\gamma$  .
Bregman Proximal Operator Considering the Burg's entropy (22) in Proposition 1 we get that, if  $\eta_{\gamma}: x \to \psi_{\gamma} \circ \nabla h^{*}(x) = \psi_{\gamma}(-\frac{1}{x})$  is convex on  $\mathrm{dom}(h^{*}) = \mathbb{R}_{- - }$ , the Bregman Score Denoiser (25) satisfies  $\mathcal{B}_{\gamma}(y) = \operatorname{Prox}_{\phi_{\gamma}}^{h}$ , for the nonconvex potential  $\psi_{\gamma}(y) = \sum_{i=1}^{n} \log(y_i) - g_{\gamma}(y) - 1$ . The convexity of  $\psi_{\gamma} \circ \nabla h^{*}$  can be verified using the following characterization, which translates to a condition on the deep potential  $g_{\gamma}$  (see Appendix E.2 for details)
$$
\forall x \in \operatorname {d o m} \left(h ^ {*}\right), \forall d \in \mathbb {R} ^ {n}, \langle \nabla^ {2} \eta_ {\gamma} (x) d, d \rangle \geq 0 \tag {28}
$$
$$
\Leftrightarrow \forall y \in \mathbb {R} _ {+ +} ^ {n}, \forall d \in \mathbb {R} ^ {n}, \langle y ^ {4} \nabla^ {2} g _ {\gamma} (y) d, d \rangle \leq \sum_ {i = 1} ^ {n} \left(y ^ {2} \left(1 - 2 y \nabla g _ {\gamma} (y)\right)\right) _ {i} d _ {i} ^ {2} \tag {29}
$$
(b) Noisy (17.85 dB)
Figure 1: Denoising of a  $256 \times 256$  image corrupted with Inverse Gamma noise of level  $\gamma = 25$ .
(c) DRUNet (32.56 dB)
(d) B-DRUNet (32.56 dB)
It is difficult to constrain  $g_{\gamma}$  to satisfy the above condition. In Appendix E.2, we proceed to a detailed experimental validation of this assumption. We empirically observe on images that the trained deep potential  $g_{\gamma}$  satisfies the above inequality. This suggests that the convexity condition for Proposition 1 is true at least locally, on the image manifold. Indeed, we prove in Appendix E.3 that the convexity condition holds for the theoretical MMSE denoiser (24). As explained below, our denoiser is trained by minimizing the  $L^2$  cost for which the optimal Bayes estimator is the MMSE. The MMSE is then a theoretical denoiser that our denoiser tries to approximate. This explains why our denoiser, after training, naturally satisfies this condition without necessitating supplementary constraints.
5.2 Bregman Plug-and-Play for Poisson Image Deblurring
We now derive the explicit B-RED and B-PnP algorithms in the context of Poisson image restoration. Choosing  $C = [0, R]^n$  for some  $R > 0$ , the B-RED and B-PnP algorithms (18) and (20) write
$$
\begin{array}{l} \left(\mathbf {B} - \mathbf {R E D}\right) x _ {i} ^ {k + 1} = \arg \min  \left\{x \nabla F _ {\lambda , \gamma} \left(x ^ {k}\right) _ {i} + \frac {1}{\tau} \left(\frac {x}{x _ {i} ^ {k}} - \log \frac {x}{x _ {i} ^ {k}}\right): x \in [ 0, R ] \right\} \quad 1 \leq i \leq n (30) \\ = \left\{ \begin{array}{l} \frac {x _ {i} ^ {k}}{1 + \tau x _ {i} ^ {k} \nabla F _ {\lambda , \gamma} (x ^ {k}) _ {i}} \text {i f} 0 \leq \frac {x _ {i} ^ {k}}{1 + \tau x _ {i} ^ {k} \nabla F _ {\lambda , \gamma} (x ^ {k}) _ {i}} \leq R \\ R \text {e l s e} \end{array} \quad 1 \leq i \leq n \right. (31) \\ \end{array}
$$
$$
\left(\mathbf {B} - \mathbf {P n P}\right) x ^ {k + 1} = \mathcal {B} _ {\gamma} \left(\frac {x ^ {k}}{1 + \tau x ^ {k} \nabla f \left(x ^ {k}\right)}\right). \tag {32}
$$
Verification of the assumptions of Theorems 1 and 2 for the convergence of both algorithms is discussed in Appendix E.4.
Poisson deblurring Equipped with the Bregman Score Denoiser, we now investigate the practical performance of the plug-and-play B-RED and B-PnP algorithms for image deblurring with Poisson noise. In this context, the degradation operator  $A$  is a convolution with a blur kernel. We verify the efficiency of both algorithms over a variety of blur kernels (real-world camera shake, uniform and Gaussian). The hyper-parameters  $\gamma$ ,  $\lambda$  are optimized for each algorithm and for each noise level  $\alpha$  by grid search and are given in Appendix E.5. In practice, we first initialize the algorithm with 100 steps with large  $\tau$  and  $\gamma$  so as to quickly initialize the algorithm closer to the right stationary point.
Note that the constraint  $\lambda L_{f} < 1$  for convergence of the B-PnP algorithm (Theorem 1) may not be respected. The global NoLip constant  $L_{f}$  can indeed be locally very lose. As explained in the Appendix (32), we can adopt a backtracking-like strategy on the regularization parameter  $\lambda$  to ensure convergence. Nevertheless, with the proposed default value of  $\lambda$ , this backtracking algorithm was never activated over the variety of blur kernels and noise levels experimented.
We show in Figures 2 and 3 that both B-PnP and B-RED algorithms provide good visual reconstruction. Moreover, we observe that, in practice, both algorithms satisfy the sufficient decrease property of the objective function as well as the convergence of the iterates. Additional quantitative performance and comparisons with other Poisson deblurring methods are given in Appendix E.4.
In this paper, we derive a complete extension of the plug-and-play framework in the general Bregman paradigm for non-smooth image inverse problems. Given a convex potential  $h$  adapted to the geometry
(d) B-PnP (22.96dB)
(b) Observed (14.91dB)
(e)  $(\lambda f + g_{\gamma})(x_k)$  B-RED
(c) B-RED (23.01dB)
(f)  $(\lambda f + \phi_{\gamma})(x_k)$  B-PnP
(g)  $||x_{i + 1} - x_i||^2$
Figure 2: Deblurring from the indicated motion kernel and Poisson noise with  $\alpha = 40$ .
(a) Clean
(b) Observed (16.21dB)
(e)  $(\lambda f + g_{\gamma})(x_k)$  B-RED
(c) B-RED (24.50dB)
(f)  $(\lambda f + \phi_{\gamma})(x_k)$  B-PnP
(d) B-PnP (24.51dB)
(g)  $||x_{i + 1} - x_i||^2$
Figure 3: Deblurring from the indicated Gaussian blur kernel and Poisson noise with  $\alpha = 60$ .
of the problem, we propose a new deep denoiser, parametrized by  $h$ , which provably writes as the Bregman proximal operator of a nonconvex potential. We argue that this denoiser should be trained on a particular noise model, called Bregman noise, that also depends on  $h$ . By plugging this denoiser in the BPG algorithm, we propose two new plug-and-play algorithms, called B-PnP and B-RED, and show that both algorithms converge to stationary points of explicit nonconvex functionals. We apply this framework to Poisson image inverse problem. Experiments on image deblurring illustrate numerically the convergence and the efficiency of the approach.
The central significance of our work stems from its theoretical study but we recognize certain limits within our experimental results. First, when applied to deblurring with Poisson noise, our proposed algorithms do not outperform existing methods in terms of PSNR. Second, while we prove that B-RED is convergent without restriction, the convergence of B-PnP depends on a specific convexity condition. Despite being confirmed with experiments and having robust theoretical foundations, this assumption could potentially be not verified when applied to non-natural images that significantly differ from those in the training dataset. Finally, due to the nonconvex nature of our proposed prior, the practical performance of the algorithms can be sensitive to their initialization.
Acknowledgements This work was funded by the French ministry of research through a CDSN grant of ENS Paris-Saclay. It has also been carried out with financial support from the French Research Agency through the PostProdLEAP and Mistic projects (ANR-19-CE23-0027-01 and ANR-19-CE40-005). It has also been supported by the NSF CAREER award under grant CCF-2043134.
Matthieu Terris, Audrey Repetti, Jean-Christophe Pesquet, and Yves Wiaux. Building firmly nonexpansive convolutional neural networks. In ICASSP 2020-2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 8658-8662. IEEE, 2020.
Singanallur V Venkatakrishnan, Charles A Bouman, and Brendt Wohlberg. Plug-and-play priors for model based reconstruction. In 2013 IEEE Global Conference on Signal and Information Processing, pages 945-948. IEEE, 2013.
Kai Zhang, Wangmeng Zuo, Shuhang Gu, and Lei Zhang. Learning deep cnn denoiser prior for image restoration. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 3929-3938, 2017.
Kai Zhang, Yawei Li, Wangmeng Zuo, Lei Zhang, Luc Van Gool, and Radu Timofte. Plug-and-play image restoration with deep denoiser prior. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2021.
A Definitions
A.1 Legendre functions, Bregman divergence
Definition 1 (Legendre function, Rockafellar [1997]). Let  $h: C \subseteq \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$  be a proper lower semi-continuous convex function. It is called:
(i) essentially smooth, if  $h$  is differentiable on  $\operatorname{int} \operatorname{dom}(h)$ , with moreover  $||\nabla h(x^k)|| \to \infty$  for every sequence  $\{x^k\}_{k \in \mathbb{N}}$  of  $\operatorname{int} \operatorname{dom}(h)$  converging towards a boundary point of  $\operatorname{dom}(h)$ .
(ii) of Legendre type if  $h$  is essentially smooth and strictly convex on  $\operatorname{int} \operatorname{dom}(h)$ .
We also recall the following properties of Legendre functions that are used without justification in our analysis (see [Rockafellar, 1997, Section 26] for more details):
-  $h$  is of Legendre type if and only if its convex conjugate  $h^*$  is of Legendre type.
- For  $h$  of Legendre type,  $\operatorname{dom}(\nabla h) = \operatorname{int}\operatorname{dom}(h)$ ,  $\nabla h$  is a bijection from  $\operatorname{int}\operatorname{dom}(h)$  to  $\operatorname{int}\operatorname{dom}(h^{*})$  and  $(\nabla h)^{-1} = \nabla h^{*}$ .
Examples We are interested in the two following Legendre functions
-  $L_{2}$  potential:  $h(x) = \frac{1}{2} ||x||^{2}, C = \mathrm{dom}(h) = \mathrm{dom}(h^{*}) = \mathbb{R}^{n}, D_{h}(x,y) = \frac{1}{2} ||x - y||^{2}$
- Burg's entropy:  $h(x) = -\sum_{i=1}^{n} \log(x_i)$ ,  $C = \operatorname{dom}(h) = \mathbb{R}_{++}^n$ ,  $\operatorname{dom}(h^*) = \mathbb{R}_{--}^n$  and  $D_h(x, y) = \sum_{i=1}^{n} \frac{x_i}{y_i} - \log\left(\frac{x_i}{y_i}\right) - 1$ .
A.2 Nonconvex subdifferential
Following Attouch et al. [2013], we use as notion of subdifferential of a proper, nonconvex function  $\Phi$  the limiting subdifferential
$$
\partial \Phi (x) = \left\{\omega \in \mathbb {R} ^ {n}, \exists x _ {k} \rightarrow x, \Phi \left(x _ {k}\right)\rightarrow \Phi (x), \omega_ {k} \rightarrow \omega , \omega_ {k} \in \hat {\partial} \Phi \left(x _ {k}\right)\right\} \tag {33}
$$
with  $\hat{\partial}\Phi$  the Fréchet subdifferential of  $\Phi$ . We have
$$
\left\{\omega \in \mathbb {R} ^ {n}, \exists x _ {k} \rightarrow x, \Phi \left(x _ {k}\right)\rightarrow \Phi (x), \omega_ {k} \rightarrow \omega , \omega_ {k} \in \partial \Phi \left(x _ {k}\right)\right\} \subseteq \partial \Phi (x). \tag {34}
$$
A.3 Kurdyka-Lojasiewicz (KL) property and subanalytic functions.
Definition 2 (Kurdyka-Lojasiewicz (KL) [Attouch et al., 2013]). A function  $f: \mathbb{R}^n \longrightarrow \mathbb{R} \cup +\infty$  is said to have the Kurdyka-Lojasiewicz property at  $x^{*} \in \operatorname{dom}(f)$  if there exists  $\eta \in (0, +\infty)$ , a neighborhood  $U$  of  $x^{*}$  and a continuous concave function  $\psi: [0, \eta) \longrightarrow \mathbb{R}_{+}$  such that  $\psi(0) = 0$ ,  $\psi$  is  $C^1$  on  $(0, \eta)$ ,  $\psi' > 0$  on  $(0, \eta)$  and  $\forall x \in U \cap [f(x^{*}) < f < f(x^{*}) + \eta]$ , the Kurdyka-Lojasiewicz inequality holds:
$$
\psi^ {\prime} \left(f (x) - f \left(x ^ {*}\right)\right) d i s t (0, \partial f (x)) \geq 1. \tag {35}
$$
Proper lower semicontinuous functions which satisfy the Kurdyka-Lojasiewicz inequality at each point of  $\operatorname{dom}(\partial f)$  are called KL functions.
This condition can be interpreted as the fact that, up to a reparameterization, the function is locally sharp around its critical points i.e. we can bound its subgradients away from 0. For more details and interpretations, we refer to [Attouch et al., 2010] and [Bolte et al., 2010]. As shown in Bolte et al. [2007], a large class of functions that have the KL-property is given by continuous subanalytic functions.
Definition 3 (Subanalytic [Bolte et al., 2007]).