SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Weijie Gan
Washington University in St. Louis weijie.gan@wustl.edu
Shirin Shoushtari
Washington University in St. Louis s.shirin@wustl.edu
Yuyang Hu
Washington University in St. Louis h.yuyang@wustl.edu
Jiaming Liu
Washington University in St. Louis jiaming.liu@wustl.edu
Hongyu An
Washington University in St. Louis hongyuan@wustl.edu
Ulugbek S. Kamilov
Washington University in St. Louis kamilov@wustl.edu
Plug-and-play (PnP) prior is a well-known class of methods for solving imaging inverse problems by computing fixed-points of operators combining physical measurement models and learned image denoisers. While PnP methods have been extensively used for image recovery with known measurement operators, there is little work on PnP for solving blind inverse problems. We address this gap by presenting a new block-coordinate PnP (BC-PnP) method that efficiently solves this joint estimation problem by introducing learned denoisers as priors on both the unknown image and the unknown measurement operator. We present a new convergence theory for BC-PnP compatible with blind inverse problems by considering nonconvex data-fidelity terms and expansive denoisers. Our theory analyzes the convergence of BC-PnP to a stationary point of an implicit function associated with an approximate minimum mean-squared error (MMSE) denoiser. We numerically validate our method on two blind inverse problems: automatic coil sensitivity estimation in magnetic resonance imaging (MRI) and blind image deblurring. Our results show that BC-PnP provides an efficient and principled framework for using denoisers as PnP priors for jointly estimating measurement operators and images.
Many problems in computational imaging, biomedical imaging, and computer vision can be formulated as inverse problems involving the recovery of high-quality images from low-quality observations. Imaging inverse problems are generally ill-posed, which means that multiple plausible clean images could lead to the same observation. It is thus common to introduce prior models on the desired images. While the literature on prior modeling of images is vast, current methods are often based on deep learning (DL), where a deep model is trained to map observations to images [1-3].
Plug-and-play (PnP) priors [4,5] is one of the most widely-used DL frameworks for solving imaging inverse problems. PnP methods circumvent the need to explicitly describe the full probability density of images by specifying image priors using image denoisers. The integration of state-of-the-art deep denoisers with physical measurement models within PnP has been shown to be effective in a number of inverse problems, including image super-resolution, phase retrieval, microscopy, and
medical imaging [6-13] (see also recent reviews [14, 15]). Practical success of PnP has also motivated novel extensions, theoretical analyses, statistical interpretations, as well as connections to related approaches such as score matching and diffusion models [16-29].
Despite the rich literature on PnP, the existing work on the topic has primarily focused on the problem of image recovery where the measurement operator is known exactly. There is little work on PnP for blind inverse problems, where both the image and the measurement operator are unknown. This form of blind inverse problems are ubiquitous in computational imaging with well-known applications such as blind deblurring [30] and parallel magnetic resonance imaging (MRI) [31]. In this paper, we address this gap by developing a new PnP approach that uses denoisers as priors over both the unknown measurement model and the unknown image, and efficiently solves the joint estimation task as a block-coordinate  $PnP$  ( $BC-PnP$ ) method. While a variant of BC-PnP was proposed in the recent paper [21], it was never used for jointly estimating the images and the measurement operators. Additionally, the convergence theory in [21] is inadequate for blind inverse problems since it assumes convex data-fidelity terms and nonexpansive denoisers. We present a new convergence analysis applicable to nonconvex data-fidelity terms and expansive denoisers. Our theoretical analysis provides explicit error bounds on the convergence of BC-PnP for approximate minimum mean squared error (MMSE) denoisers under a set of clearly specified assumptions. We show the practical relevance of BC-PnP by solving joint estimation problems in blind deblurring and accelerated parallel MRI. Our numerical results show the potential of denoisers to act as PnP priors over the measurement operators as well as images. Our work thus addresses a gap in the current PnP literature by providing a new efficient and principled framework applicable to a wide variety of blind imaging inverse problems.
All proofs and some details that have been omitted for space appear in the supplementary material.
Inverse Problems. Many imaging problems can be formulated as inverse problems where the goal is to estimate an unknown image  $\pmb{x} \in \mathbb{R}^n$  from its degraded observation  $\pmb{y} = \pmb{A}\pmb{x} + \pmb{e}$ , where  $\pmb{A} \in \mathbb{R}^{m \times n}$  is a measurement operator and  $\pmb{e} \in \mathbb{R}^m$  is the noise. A common approach for solving inverse problems is based on formulating an optimization problem
$$
\widehat {\boldsymbol {x}} \in \underset {\boldsymbol {x} \in \mathbb {R} ^ {n}} {\arg \min } f (\boldsymbol {x}) \quad \text {w i t h} \quad f (\boldsymbol {x}) = g (\boldsymbol {x}) + h (\boldsymbol {x}), \tag {1}
$$
where  $g$  is the data-fidelity term that quantifies consistency with the observation  $\mathbf{y}$  and  $h$  is the regularizer that infuses a prior on  $\mathbf{x}$ . For example, a widely-used data-fidelity term and regularizer in computational imaging are the least-squares  $g(\mathbf{x}) = \frac{1}{2}\| \mathbf{A}\mathbf{x} - \mathbf{y}\|_2^2$  and the total variation (TV) functions  $h(\mathbf{x}) = \tau \| \mathbf{D}\mathbf{x}\|_1$ , where  $\mathbf{D}$  is the image gradient, and  $\tau > 0$  a regularization parameter.
The traditional inverse problem formulations assume that the measurement operator  $A$  is known exactly. However, in many applications, it is more practical to model the measurement operator as  $A(\theta)$ , where  $\theta \in \mathbb{R}^p$  are unknown parameters to be estimated jointly with  $x$ . This form of inverse problems are often referred to as blind inverse problems and arise in a wide-variety of applications, including prallel MRI [32-37], blind deblurring [30, 38, 39], and computed tomography [40-43].
DL. There is a growing interest in DL for solving imaging inverse problems [1-3]. Instead of explicitly defining a regularizer, DL approaches for solving inverse problems learn a mapping from the measurements to the desired image by training a convolutional neural network (CNN) to perform regularized inversion [44-48]. Model-based DL (MBDL) has emerged as powerful DL framework for inverse problems that combines the knowledge of the measurement operator with an image prior specified by a CNN (see reviews [3, 49]). The literature of MBDL is vast, but some well-known examples include PnP, regularization by denoising (RED), deep unfolding (DU), compressed sensing using generative models (CSGM), and deep equilibrium models (DEQ) [50-54]. All these approaches come with different trade-offs in terms of imaging performance, computational and memory complexity, flexibility, need for supervision, and theoretical understanding.
The literature on DL approaches for blind inverse problems is broad, with many specialized methods developed for different applications. While an in-depth review would be impractical for this paper, we mention several representative approaches adopted in prior work. The direct application of DL to predict the measurement operator from the observation was explored in [55, 56]. Deep image prior (DIP) was used as a learning-free prior to regularize the image and the measurement operator
in [57, 58]. Generative models, including both GANs and diffusion models, have been explored as regularizers for blind inverse problems in [59, 60]. Other work considered the use of a dedicated neural network to predict the parameters of the measurement operator, adoption of model adaptation strategies, and development of autocalibration methods based on optimization [34-37, 61-64].
PnP. PnP [4, 5] is one of the most popular MBDL approaches based on using deep denoisers as imaging priors (see also recent reviews [14, 15]). For example, the proximal gradient method variant of PnP (referred to as PnP-ISTA in this paper) can be formulated as a fixed-point iteration [65]
$$
\boldsymbol {x} ^ {k} \leftarrow \mathrm {D} _ {\sigma} \left(\boldsymbol {z} ^ {k}\right) \quad \text {w i t h} \quad \boldsymbol {z} ^ {k} \leftarrow \boldsymbol {x} ^ {k - 1} - \gamma \nabla g (\boldsymbol {x} ^ {k - 1}), \tag {2}
$$
where  $\mathrm{D}_{\sigma}$  is a denoiser with a parameter  $\sigma > 0$  for controlling its strength and  $\gamma > 0$  is a step-size. The theoretical convergence of PnP-ISTA has been explored for convex functions  $g$  using monotone operator theory [20, 22] as well as for nonconvex functions based on interpreting the denoiser as a MMSE estimator [23]. The analysis in this paper builds on the convergence theory in [23] that uses an elegant formulation by Gribonval [66] establishing a direct link between MMSE estimation and regularized inversion. Many variants of PnP have been developed over the past few years [6-12], which has motivated an extensive research on its theoretical properties [16, 18, 20, 22, 23, 27-29, 67-70].
Block coordinate regularization by denoising (BC-RED) is a recent PnP variant for solving large-scale inverse problems by updating along a subset of coordinates at every iteration [21]. BC-RED is based on regularization by denoising (RED), another well-known variant of PnP that seeks to formulate an explicit regularizer for a given image denoiser [17, 19]. BC-RED was applied to several non-blind inverse problems and was theoretically analyzed for convex data-fidelity terms.
PnP was extended to blind deblurring in [71, 72] by considering an additional prior on blur kernels that promotes sparse and nonenegative solutions. PnP was also applied to holography with unknown phase errors by using the Gaussian Markov random field model as the prior for the phase errors [73]. Calibrated RED (Cal-RED) [43] is a recent related extension of RED that calibrates the measurement operator during RED reconstruction by combining the traditional RED updates over an unknown image with a gradient descent over the unknown parameters of the measurement operator. However, this prior work does not leverage any learned priors for the measurement operator and does not provide any theoretical analysis.
Our contributions. (1) Our first contribution is in the use of learned deep denoisers for regularizing the measurement operators within PnP. While the idea of calibration within PnP was introduced in [43], denoisers were not used as priors for measurement operators. (2) Our second contribution is the application of BC-PnP as an efficient method for jointly estimating the unknown image and the measurement operator. While BC-RED was introduced in [21] as a block-coordinate variant of PnP, the method was used for solving non-blind inverse problems by using patch-based image denoisers. (3) Our third contribution is a new convergence theory for BC-PnP for the sequential and random block-selection strategies under approximate MMSE denoisers. Our analysis does not assume convex data-fidelity terms, which makes it compatible with blind inverse problems. Our analysis can be seen as an extension of [23] to block-coordinate updates and approximate MMSE denoisers. (4) Our fourth contribution is the implementation of BC-PnP using learned deep denoisers as priors for two distinct blind inverse problems: blind deblurring and auto-calibrated parallel MRI. Our code—which we share publicly—shows the potential of learning deep denoisers over measurement operators and using them for jointly estimating the unknown image and the unknown measurement operator.
3 Block Coordinate Plug-and-Play Method
We propose to efficiently solve blind inverse problems by using a block-coordinate PnP method, where each block represents one group of unknown variables (images, measurement operators, etc). The novelty of our work relative to [21] is in solving blind inverse problems by introducing learned priors on both the unknown image and the unknown measurement operator. Additionally, unlike [21], our work proposes a fully nonconvex formulation that is more applicable to blind inverse problems.
Consider the decomposition of a vector  $\pmb{x} \in \mathbb{R}^n$  into  $b \geq 1$  blocks
$$
\boldsymbol {x} = \left(\boldsymbol {x} _ {1}, \dots , \boldsymbol {x} _ {b}\right) \in \mathbb {R} ^ {n _ {1}} \times \dots \times \mathbb {R} ^ {n _ {b}} \quad \text {w i t h} \quad n = n _ {1} + \dots + n _ {b}. \tag {3}
$$
Algorithm 1 Block Coordinate Plug-and-Play Method (BC-PnP)
1: input: initial value  $\pmb{x}^0 \in \mathbb{R}^n$ , parameters  $\pmb{\sigma} \in \mathbb{R}_{+}^{b}$ , and step-size  $\gamma > 0$ .
2: for  $k = 1,2,3,\dots$  do
3: Choose an index  $i_k \in \{1, \dots, b\}$
4:  $\pmb{x}^k\gets \pmb{x}^{k - 1} - \gamma \mathbf{U}_{i_k}\mathbf{G}_{i_k}(\pmb{x}^{k - 1})$
where  $\mathsf{G}_i(\pmb {x})\coloneqq \mathbf{U}_i^\top \mathsf{G}(\pmb {x})$  with  $\mathsf{G}(\pmb {x})\coloneqq \frac{1}{\gamma} (\pmb {x} - \mathsf{D}_{\pmb{\sigma}}(\pmb {x} - \gamma \nabla g(\pmb {x}))).$
5: end for
For each  $i \in \{1, \dots, b\}$ , we define a matrix  $\mathbf{U}_i \in \mathbb{R}^{n \times n_i}$  that injects a vector in  $\mathbb{R}^{n_i}$  into  $\mathbb{R}^n$  and its transpose  $\mathbf{U}_i^\top$  that extracts the  $i$ th block from a vector in  $\mathbb{R}^n$ . For any  $\boldsymbol{x} \in \mathbb{R}^n$ , we have
$$
\boldsymbol {x} = \sum_ {i = 1} ^ {b} \mathbf {U} _ {i} \boldsymbol {x} _ {i} \quad \text {w i t h} \quad \boldsymbol {x} _ {i} = \mathbf {U} _ {i} ^ {\top} \boldsymbol {x} \in \mathbb {R} ^ {n _ {i}}, \quad i = 1, \dots , b \quad \Leftrightarrow \quad \sum_ {i = 1} ^ {b} \mathbf {U} _ {i} \mathbf {U} _ {i} ^ {\top} = \mathbf {I}. \tag {4}
$$
Note that (4) directly implies the norm preservation  $\| \pmb{x}\| _2^2 = \| \pmb{x}_1\| _2^2 +\dots +\| \pmb{x}_b\| _2^2$  for any  $\pmb {x}\in \mathbb{R}^n$ . We are interested in a block-coordinate algorithm that uses only a subset of operator outputs corresponding to coordinates in some block  $i\in \{1,\dots ,b\}$ . Hence, for an operator  $\mathsf{G}:\mathbb{R}^n\to \mathbb{R}^n$ , we define the block-coordinate operator  $\mathsf{G}_i:\mathbb{R}^n\to \mathbb{R}^{n_i}$  as
$$
\mathbf {G} _ {i} (\boldsymbol {x}) := \left[ \mathbf {G} (\boldsymbol {x}) \right] _ {i} = \mathbf {U} _ {i} ^ {\mathrm {T}} \mathbf {G} (\boldsymbol {x}) \in \mathbb {R} ^ {n _ {i}}, \quad \boldsymbol {x} \in \mathbb {R} ^ {n}. \tag {5}
$$
We are in-particular interested in two operators: (a) the gradient  $\nabla g(\pmb{x}) = (\nabla_1g(\pmb{x}),\dots ,\nabla_bg(\pmb{x}))$  of the data-fidelity term  $g$  and (b) the denoiser  $\mathsf{D}_{\pmb{\sigma}}(\pmb{x}) = (\mathsf{D}_{\sigma_1}(\pmb{x}_1),\dots ,\mathsf{D}_{\sigma_b}(\pmb{x}_b))$ , where the vector  $\pmb{\sigma} = (\sigma_{1},\dots,\sigma_{b})\in \mathbb{R}_{+}^{b}$  consists of parameters for controlling the strength of each block denoiser. Note how the denoiser acts in a separable fashion across different blocks.
When  $b = 1$ , we have  $\mathbf{U}_1 = \mathbf{U}_1^\top = \mathbf{I}$  and BC-PnP reduces to the conventional PnP-ISTA [23,65]. When  $b > 1$ , we have at least two blocks with BC-PnP updating only one block at a time
$$
\boldsymbol {x} _ {j} ^ {k} = \left\{ \begin{array}{l l} \boldsymbol {x} _ {j} ^ {k - 1} & \text {w h e n} j \neq i _ {k} \\ \mathrm {D} _ {\sigma_ {j}} \left(\boldsymbol {x} _ {j} ^ {k - 1} - \gamma \nabla_ {j} g \left(\boldsymbol {x} ^ {k - 1}\right)\right) & \text {w h e n} j = i _ {k}, \quad j \in \{1, \dots , b \}. \end{array} \right. \tag {6}
$$
As with any coordinate descent method (see [74] for a review), BC-PnP can be implemented using different block selection strategies. One common strategy is to simply update blocks sequentially as  $i_k = 1 + \bmod(k - 1, b)$ , where  $\bmod(\cdot)$  denotes the modulo operator. An alternative is to proceed in epochs of  $b$  consecutive iterations, where at the start of each epoch the set  $\{1, \dots, b\}$  is reshuffled, and  $i_k$  is then selected consecutively from this ordered set. Finally, one can adopt a fully randomized strategy where indices  $i_k$  are selected as i.i.d. random variables distributed uniformly over  $\{1, \dots, b\}$ .
Throughout this work, we will assume that each denoiser  $\mathsf{D}_{\sigma_i}$  is an approximate MMSE estimator for the following AWGN denoising problem
$$
\boldsymbol {z} _ {i} = \boldsymbol {x} _ {i} + \boldsymbol {n} _ {i} \quad \text {w i t h} \quad \boldsymbol {x} _ {i} \sim p _ {\boldsymbol {x} _ {i}}, \quad \boldsymbol {n} _ {i} \sim \mathcal {N} (0, \sigma_ {i} ^ {2} \mathbf {I}), \tag {7}
$$
where  $i \in \{1, \dots, b\}$  and  $\pmb{z}_i \in \mathbb{R}^{n_i}$ . We rely only on an approximation of the MMSE estimator of  $\pmb{x}_i$  given  $\pmb{z}_i$ , since the exact MMSE denoiser corresponds to the generally intractable posterior mean
$$
\mathrm {D} _ {\sigma_ {i}} ^ {*} (\boldsymbol {z} _ {i}) := \mathbb {E} [ \boldsymbol {x} _ {i} | \boldsymbol {z} _ {i} ] = \int_ {\mathbb {R} ^ {n _ {i}}} \boldsymbol {x} p _ {\boldsymbol {x} _ {i} | \boldsymbol {z} _ {i}} (\boldsymbol {x} | \boldsymbol {z} _ {i}) \mathrm {d} \boldsymbol {x}. \tag {8}
$$
Approximate MMSE denoisers are a useful model for denoisers due to the use of the MSE loss
$$
\mathcal {L} \left(\mathrm {D} _ {\sigma_ {i}}\right) = \mathbb {E} \left[ \left\| \boldsymbol {x} _ {i} - \mathrm {D} _ {\sigma_ {i}} \left(\boldsymbol {z} _ {i}\right) \right\| _ {2} ^ {2} \right] \tag {9}
$$
for training deep denoisers, as well as the optimality of MMSE denoisers with respect to widely used image-quality metrics such as signal-to-noise ratio (SNR).
As a simple illustration of the generality of BC-PnP, consider  $b = 2$  with the least-squares objective
$$
g (\boldsymbol {x}) = \frac {1}{2} \| \boldsymbol {y} - \boldsymbol {A} (\boldsymbol {\theta}) \boldsymbol {v} \| _ {2} ^ {2} \quad \text {w i t h} \quad \boldsymbol {x} := (\boldsymbol {v}, \boldsymbol {\theta}), \tag {10}
$$
where  $\pmb{v} \in \mathbb{R}^{n_1}$  denotes the unknown image and  $\pmb{\theta} \in \mathbb{R}^{n_2}$  denotes the unknown parameters of the measurement operator. BC-PnP can then be implemented by first pre-training a dedicated AWGN denoiser  $\mathsf{D}_{\sigma_i}$  for each block  $i$  and using it as a prior within Algorithm 1. It is also worth noting that the functions  $g$  in (10) is nonconvex with respect to the variable  $\pmb{x} \in \mathbb{R}^n$ . In the next section, we present the full convergence analysis of BC-PnP without any convexity assumptions on  $g$  and nonexpansiveness assumptions on the denoiser  $\mathsf{D}_{\sigma}$ .
4 Convergence Analysis of BC-PnP
In this section, we present two new theoretical convergence results for BC-PnP. We first discuss its convergence under the sequential updates and then under fully random updates. It is worth mentioning that BC-RED with fully random updates was theoretically analyzed in [21]. The novelty of our analysis here lies in that it allows for nonconvex functions  $g$  and expansive denoisers  $\mathsf{D}_{\sigma_i}$ . The nonconvexity of  $g$  is essential since most data-fidelity terms used for blind inverse problems are nonconvex. On the other hand, by allowing expansive  $\mathsf{D}_{\sigma_i}$ , our analysis avoids the need for the spectral normalization techniques that were previously suggested for PnP methods [21, 22].
In the following, we will denote as  $\mathsf{D}_{\sigma}^{*} := (\mathsf{D}_{\sigma_{1}}^{*}, \dots, \mathsf{D}_{\sigma_{b}}^{*})$  the exact MMSE denoiser in (8). Our analysis will require five assumptions that will serve as sufficient conditions for our theorems.
Assumption 1. The blocks  $\pmb{x}_i$  are independent with non-degenerate priors  $p_{\pmb{x}_i}$  over  $\mathbb{R}^{n_i}$ .
As a reminder, a probability distribution  $p_{\pmb{x}_i}$  is degenerate over  $\mathbb{R}^{n_i}$ , if it is supported on a space of lower dimensions than  $n_i$ . Assumption 1 is required for establishing an explicit link between the MMSE denoiser (8) and the following regularizer (see also [23, 66] for additional background)
$$
h (\boldsymbol {x}) = \sum_ {i = 1} ^ {b} h _ {i} \left(\boldsymbol {x} _ {i}\right), \quad \boldsymbol {x} = \left(\boldsymbol {x} _ {1}, \dots , \boldsymbol {x} _ {n}\right) \in \mathbb {R} ^ {n}, \tag {11}
$$
where each function  $h_i$  is defined as (see the derivation in Section D.2 of the supplement)
$$
h _ {i} \left(\boldsymbol {x} _ {i}\right) := \left\{ \begin{array}{l l} - \frac {1}{2 \gamma} \| \boldsymbol {x} _ {i} - \left(\mathrm {D} _ {\sigma_ {i}} ^ {*}\right) ^ {- 1} \left(\boldsymbol {x} _ {i}\right) \| _ {2} ^ {2} + \frac {\sigma_ {i} ^ {2}}{\gamma} h _ {\sigma_ {i}} \left(\left(\mathrm {D} _ {\sigma_ {i}} ^ {*}\right) ^ {- 1} \left(\boldsymbol {x} _ {i}\right)\right) & \text {f o r} \boldsymbol {x} _ {i} \in \operatorname {I m} \left(\mathrm {D} _ {\sigma_ {i}} ^ {*}\right) \\ + \infty & \text {f o r} \boldsymbol {x} _ {i} \notin \operatorname {I m} \left(\mathrm {D} _ {\sigma_ {i}} ^ {*}\right), \end{array} \right. \tag {12}
$$
where  $\gamma > 0$  is the step size,  $(\mathsf{D}_{\sigma_i}^*)^{-1} : \mathsf{Im}(\mathsf{D}_{\sigma_i}^*) \to \mathbb{R}^{n_i}$  is the inverse mapping, which is well defined and smooth over  $\mathsf{Im}(\mathsf{D}_{\sigma_i}^*)$ , and  $h_{\sigma_i}(\cdot) := -\log(p_{\pmb{z}_i}(\cdot))$ , where  $p_{\pmb{z}_i}$  is the probability distribution over the AWGN corrupted observations (7). Note that the function  $h_i$  is smooth for any  $\pmb{x}_i \in \mathsf{Im}(\mathsf{D}_{\sigma_i}^*)$ , which is the consequence of the smoothness of both  $(\mathsf{D}_{\sigma_i}^*)^{-1}$  and  $h_{\sigma_i}$ .
Assumption 2. The function  $g$  is continuously differentiable and  $\nabla g$  is Lipschitz continuous with constant  $L > 0$ . Additionally, each block gradient  $\nabla_{i}g$  is block Lipschitz continuous with constant  $L_{i} > 0$ . We define the largest block Lipschitz constant as  $L_{\max} \coloneqq \max \{L_1,\dots ,L_b\}$ .
Lipschitz continuity of the gradient  $\nabla g$  is a standard assumption in the context of imaging inverse problems. Note that we always have the relationship  $(L / b)\leq L_{\max}\leq L$  (see Section 3.2 in [74]).
Assumption 3. The explicit data-fidelity term and the implicit regularizer are bounded from below
$$
\inf  _ {\boldsymbol {x} \in \mathbb {R} ^ {n}} g (\boldsymbol {x}) > - \infty , \quad \inf  _ {\boldsymbol {x} \in \mathbb {R} ^ {n}} h (\boldsymbol {x}) > - \infty . \tag {13}
$$
Assumption 3 implies that there exists  $f^{*} > -\infty$  such that  $f(\pmb{x}) \geq f^{*}$  for all  $\pmb{x} \in \mathbb{R}^n$ . Since Assumptions 1-3 correspond to the standard assumptions used in the literature, they are broadly satisfied in the context of inverse problems.
Our analysis assumes that at every iteration, BC-PnP uses inexact MMSE denoisers on each block. While there are several ways to specify the nature of "inexactness," we consider the case where at every iteration  $k$  of BC-PnP the distance of the output of  $\mathsf{D}_{\sigma_i}$  to  $\mathsf{D}_{\sigma_i}^*$  is bounded by a constant  $\varepsilon_k$ .
Assumption 4. Each block denoiser  $\mathsf{D}_{\sigma_i}$  in  $\mathsf{D}_{\sigma}$  satisfies
$$
\| \mathsf {D} _ {\sigma_ {i}} \left(\boldsymbol {z} _ {i} ^ {k}\right) - \mathsf {D} _ {\sigma_ {i}} ^ {*} \left(\boldsymbol {z} _ {i} ^ {k}\right) \| _ {2} \leq \varepsilon_ {k}, \quad i \in \{1, \dots , b \}, \quad k = 1, 2, 3, \dots ,
$$
where  $\mathsf{D}_{\sigma_i}^*$  is given in (8) and  $z_{i}^{k} = x_{i}^{k - 1} - \gamma \nabla_{i}g(\pmb{x}^{k - 1})$