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On the Strong Convexity of PnP Regularization Using Linear Denoisers

Arghya Sinha, Kunal N Chaudhury

Abstract

In the Plug-and-Play (PnP) method, a denoiser is used as a regularizer within classical proximal algorithms for image reconstruction. It is known that a broad class of linear denoisers can be expressed as the proximal operator of a convex regularizer. Consequently, the associated PnP algorithm can be linked to a convex optimization problem $\mathcal{P}$. For such a linear denoiser, we prove that $\mathcal{P}$ exhibits strong convexity for linear inverse problems. Specifically, we show that the strong convexity of $\mathcal{P}$ can be used to certify objective and iterative convergence of any PnP algorithm derived from classical proximal methods.

On the Strong Convexity of PnP Regularization Using Linear Denoisers

Abstract

In the Plug-and-Play (PnP) method, a denoiser is used as a regularizer within classical proximal algorithms for image reconstruction. It is known that a broad class of linear denoisers can be expressed as the proximal operator of a convex regularizer. Consequently, the associated PnP algorithm can be linked to a convex optimization problem . For such a linear denoiser, we prove that exhibits strong convexity for linear inverse problems. Specifically, we show that the strong convexity of can be used to certify objective and iterative convergence of any PnP algorithm derived from classical proximal methods.

Paper Structure

This paper contains 12 sections, 9 theorems, 16 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Results and Discussion
  4. Symmetric Denoisers
  5. Kernel Denoisers
  6. Numerical Results
  7. Conclusion
  8. Appendix
  9. Proof of Lemma \ref{['lemma:contractivity']}
  10. Proof of Theorem \ref{['thm:PnP-ISTA_contractivity']}
  11. Proof of Theorem \ref{['thm:PnP-ADMM_contractivity']}
  12. Proof of Theorem \ref{['thm:PnP-kernel']}

Key Result

Proposition 1

Let $\{(\boldsymbol{x}_k, \boldsymbol{y}_k, \boldsymbol{z}_k)\}_{k \geqslant 1}$ be the iterates generated by PnP-ADMM starting from $\boldsymbol{y}_0, \boldsymbol{z}_0 \in \mathbb{R}^n$. Define the sequence $\{\boldsymbol{u}_k\}_{k \geqslant 1}$ as follows: $\boldsymbol{u}_1 = \boldsymbol{y}_1 + \b for $k \geqslant 1$. Then $\boldsymbol{y}_k = \mathbf{W}\boldsymbol{u}_k$ for $k \geqslant 1$.

Figures (4)

  • Figure 1: Contraction factors for PnP-ISTA (left) and PnP-ADMM (right) for inpainting (INP), deblurring (DBR), and superresolution (SR); $\mathbf{W}_{\hbox{sym}}$ is the symmetric DSG-NLM denoiser sreehari2016plug and $\mathbf{W}_{\hbox{ker}}$ is the NLM denoiser gavaskar2021plug.
  • Figure 2: PnP-ISTA iterates for inpainting, deblurring, and superresolution, using the symmetric DSG-NLM denoiser sreehari2016plug and for different step size $\gamma$; $\boldsymbol{x}^*$ is the iterate after 100 iterations.
  • Figure 3: Inpainting (top) and $2\hbox{x}$-superresolution (bottom) using the symmetric DSG-NLM denoiser sreehari2016plug. PSNR: (a) $23.23\,dB$, (b) $30.22\,dB$, (c) $30.25\,dB$; (d) $8.40\,dB$, (e) $25.56\,dB$, (f) $25.57\,dB$.
  • Figure 4: Deblurring results for PnP-ADMM using the symmetric GMM denoiser Teodoro2019PnPfusion. PSNR for three different initializations (see insets): (a) $24.04\,dB$, (b) $24.04\,dB$, (c) $24.04\,dB$.

Theorems & Definitions (23)

  • Proposition 1
  • Proposition 2
  • Definition 1
  • Lemma 1
  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Corollary 2
  • Definition 2
  • Remark 1
  • ...and 13 more