Plug-and-Play Bregman Proximal Gradient

• The paper demonstrates that PnP–BPGM replace quadratic norms with Bregman divergences, yielding explicit convergence and efficient algorithms for inverse problems.
• The method integrates learned denoisers tuned to the problem geometry, achieving robust noise and blur removal in Poisson deblurring tasks.
• The framework generalizes traditional plug‐and‐play methods, combining rigorous theory with strong empirical performance validated by competitive PSNR metrics.

Plug-and-Play Bregman Proximal Gradient Methods (PnP–BPGM) constitute a class of optimization algorithms designed for inverse problems with non-Euclidean data-fidelity terms, particularly those arising in Poisson and other non-Gaussian noise models. By systematically replacing the quadratic norm in traditional Proximal Gradient Methods with a Bregman divergence adapted to the problem geometry, and integrating learned denoisers parametrized to this geometry, PnP–BPGM generalizes plug-and-play frameworks for a broader class of signal restoration settings. The approach is grounded in convex and nonconvex optimization theory and features explicit convergence guarantees, efficient algorithms, and strong empirical performance, especially for Poisson inverse problems (Hurault et al., 2023, Al-Shabili et al., 2022).

1. Mathematical Foundations and Bregman Geometry

The Bregman divergence $D_h(x, y)$, defined for a strictly convex, differentiable function $h: C \subseteq \mathbb{R}^n \to \mathbb{R}$, captures non-Euclidean geometry intrinsic to many inverse problems. For $h$ $\mu$-strongly convex with $L_h$-Lipschitz $\nabla h$, the divergence is

$$D_h(x, y) = h(x) - h(y) - \langle \nabla h(y),\, x-y \rangle, \quad x, y \in \operatorname{dom} h,$$

admitting lower and upper quadratic bounds: $$\frac{\mu_h}{2}\|x-y\|^2 \le D_h(x, y) \le \frac{L_h}{2}\|x-y\|^2.$$ If $h(x) = \frac12\|x\|^2$, $D_h$ specializes to the standard Euclidean norm. For Poisson noise and non-negativity constraints ($h: C \subseteq \mathbb{R}^n \to \mathbb{R}$0), Burg entropy $h: C \subseteq \mathbb{R}^n \to \mathbb{R}$1 yields $h: C \subseteq \mathbb{R}^n \to \mathbb{R}$2, closely tracking the local structure of the Kullback–Leibler data fidelity (Hurault et al., 2023, Al-Shabili et al., 2022).

2. Problem Formulation and Classical Methods

Inverse problems are formulated as minimizations of composite functionals,

$h: C \subseteq \mathbb{R}^n \to \mathbb{R}$3

where $h: C \subseteq \mathbb{R}^n \to \mathbb{R}$4 encodes the data-fidelity term—often a negative log-likelihood such as, for Poisson statistics,

$h: C \subseteq \mathbb{R}^n \to \mathbb{R}$5

with $h: C \subseteq \mathbb{R}^n \to \mathbb{R}$6—and $h: C \subseteq \mathbb{R}^n \to \mathbb{R}$7 imposes a (possibly nonconvex) regularizer.

Classical Proximal Gradient Methods (PGM) assume $h: C \subseteq \mathbb{R}^n \to \mathbb{R}$8 is $h: C \subseteq \mathbb{R}^n \to \mathbb{R}$9-smooth (Lipschitz gradient), leading to a majorize–minimize update,

$h$0

In the Plug-and-Play (PnP) paradigm, proximal operators are replaced by powerful Gaussian denoisers, yielding iterates $h$1 with $h$2. However, this Euclidean setting is ill-suited for non-Gaussian models or settings lacking efficient proximal maps (Al-Shabili et al., 2022).

3. Bregman Proximal Gradient and Plug-and-Play Integration

The Bregman Proximal Gradient Method (BPGM) generalizes PGM by using $h$3-adapted geometry. Assuming $h$4 is $h$5-smooth relative to $h$6 ($h$7 convex), the update is

$h$8

This decomposes into a mirror-descent step $h$9, followed by a Bregman proximal step $\mu$0.

Plug-and-Play BPGM (PnP–BPGM) replaces the Bregman proximal operation with a learned denoiser $\mu$1 tuned to the geometry,

$\mu$2

For Poisson models, the denoiser is parameterized to respect the Bregman noise model: $\mu$3, with $\mu$4 a learnable potential, ensuring $\mu$5 approximates the MMSE estimator under Bregman noise (Hurault et al., 2023).

4. Algorithmic Framework and Variants

Two principal algorithms are established:

• Bregman RED (B-RED): Employs a Bregman descent with backtracking and explicit update, $$D_h(x, y) = h(x) - h(y) - \langle \nabla h(y),\, x-y \rangle, \quad x, y \in \operatorname{dom} h,$$2
• Bregman PnP (B-PnP): Fixed-step implementation with plug-and-play denoising, $$D_h(x, y) = h(x) - h(y) - \langle \nabla h(y),\, x-y \rangle, \quad x, y \in \operatorname{dom} h,$$3 B-RED allows adaptive step-size adjustment; B-PnP enforces $\mu$6 for fixed-τ convergence (Hurault et al., 2023). Both leverage Burg entropy for Poisson problems but can accommodate other choices of $\mu$7 for different noise or constraints (Al-Shabili et al., 2022).

5. Theoretical Analysis: Convergence Guarantees

Convergence is established under the following:

• $\mu$8 is Legendre, $\mu$9, strongly convex;
• $L_h$0 is semialgebraic, proper, coercive, and $L_h$1 on $L_h$2;
• The “NoLip” property: $L_h$3 convex;
• $L_h$4 and $L_h$5 Lipschitz on sublevel sets;
• Learnable potentials $L_h$6, $L_h$7 lower bounded, semialgebraic.

Invoking the Kurdyka–Łojasiewicz property, the following is shown:

• The sequence $L_h$8 is nonincreasing and converges.
• Iterates $L_h$9 remain bounded in a compact sublevel set.
• $\nabla h$0, with $\nabla h$1.
• All accumulation points are stationary for the explicit global functional,
  • $\nabla h$2 for B-RED,
  • $\nabla h$3 for B-PnP (Hurault et al., 2023).

In the strongly convex case, Lipschitz-averaged denoiser conditions yield global contraction, and step-size bounds can be explicitly characterized (Al-Shabili et al., 2022).

6. Implementation and Empirical Evaluation

When specialized to Poisson deblurring, the forward model $\nabla h$4 is paired with Burg entropy, yielding closed-form mirror steps. For practical denoising, DnCNN-type architectures are trained according to Bregman noise and integrated as $\nabla h$5 (Hurault et al., 2023, Al-Shabili et al., 2022).

Experimental evaluation on CBSD68 (color images) with four blur kernels (real camera shakes, uniform $\nabla h$6, Gaussian $\nabla h$7) demonstrates:

Noise Level (α)	B-RED PSNR (dB)	B-PnP PSNR (dB)
20	23.58	23.29
40	24.54	24.54
60	24.90	24.80

Both methods exhibit monotonic decrease of $\nabla h$8 and convergence rate $\nabla h$9, achieving effective noise and blur removal without visual artifacts. In comparative studies, PnP–BPGM outperforms Euclidean PnP and direct U-Net baselines for Poisson denoising (Hurault et al., 2023, Al-Shabili et al., 2022).

7. Broader Significance and Research Directions

PnP–BPGM generalizes plug-and-play inverse problem solvers beyond the Euclidean setting, adapting the optimization geometry to the statistical structure of the data-fidelity term and the implied signal constraints. This flexibility is critical for Poisson models, nonnegativity, simplex constraints, and other scenarios where the standard quadratic proximal geometry is poorly matched. Theoretical results cover both strongly convex and certain nonconvex settings, but extensions to fully general convex/nonconvex cases, automated selection or learning of $$D_h(x, y) = h(x) - h(y) - \langle \nabla h(y),\, x-y \rangle, \quad x, y \in \operatorname{dom} h,$$0, and integration with more complex noise or signal geometries are active research topics. Limitations include the need for tractable forms of $$D_h(x, y) = h(x) - h(y) - \langle \nabla h(y),\, x-y \rangle, \quad x, y \in \operatorname{dom} h,$$1 and strong convexity assumptions in current theory (Al-Shabili et al., 2022).

PnP–BPGM thus represents a principled framework for optimizing and learning in non-Euclidean statistical inverse problems, with rigorous theoretical guarantees and practical efficacy validated in Poisson linear imaging and beyond (Hurault et al., 2023, Al-Shabili et al., 2022).