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Linear Convergence of Plug-and-Play Algorithms with Kernel Denoisers

Arghya Sinha, Bhartendu Kumar, Chirayu D. Athalye, Kunal N. Chaudhury

TL;DR

The paper advances the theory of Plug-and-Play image reconstruction by establishing global linear convergence for ISTA and ADMM when using kernel denoisers, including nonsymmetric ones, through a scaled framework and a $\mathbf{D}$-inner product that renders kernel denoisers self-adjoint. It proves contractivity of the scaled update operators under the Restricted Nullity Property and derives quantitative contraction-factor bounds that depend on the denoiser's spectral gap, the forward operator, and algorithmic parameters. Theoretical results are complemented by numerical experiments on inpainting, deblurring, and superresolution, showing that larger spectral gaps (achieved by wider kernels) accelerate convergence while highlighting a reconstruction-quality tradeoff. Overall, the work provides a unified convergence theory for symmetric and nonsymmetric kernel denoisers and offers practical guidance on denoiser design and parameter choices to achieve faster PnP convergence. The findings deepen the understanding of how kernel-based priors interact with forward-model properties to guarantee stable, linear convergence in PnP algorithms.

Abstract

The use of denoisers for image reconstruction has shown significant potential, especially for the Plug-and-Play (PnP) framework. In PnP, a powerful denoiser is used as an implicit regularizer in proximal algorithms such as ISTA and ADMM. The focus of this work is on the convergence of PnP iterates for linear inverse problems using kernel denoisers. It was shown in prior work that the update operator in standard PnP is contractive for symmetric kernel denoisers under appropriate conditions on the denoiser and the linear forward operator. Consequently, we could establish global linear convergence of the iterates using the contraction mapping theorem. In this work, we develop a unified framework to establish global linear convergence for symmetric and nonsymmetric kernel denoisers. Additionally, we derive quantitative bounds on the contraction factor (convergence rate) for inpainting, deblurring, and superresolution. We present numerical results to validate our theoretical findings.

Linear Convergence of Plug-and-Play Algorithms with Kernel Denoisers

TL;DR

The paper advances the theory of Plug-and-Play image reconstruction by establishing global linear convergence for ISTA and ADMM when using kernel denoisers, including nonsymmetric ones, through a scaled framework and a -inner product that renders kernel denoisers self-adjoint. It proves contractivity of the scaled update operators under the Restricted Nullity Property and derives quantitative contraction-factor bounds that depend on the denoiser's spectral gap, the forward operator, and algorithmic parameters. Theoretical results are complemented by numerical experiments on inpainting, deblurring, and superresolution, showing that larger spectral gaps (achieved by wider kernels) accelerate convergence while highlighting a reconstruction-quality tradeoff. Overall, the work provides a unified convergence theory for symmetric and nonsymmetric kernel denoisers and offers practical guidance on denoiser design and parameter choices to achieve faster PnP convergence. The findings deepen the understanding of how kernel-based priors interact with forward-model properties to guarantee stable, linear convergence in PnP algorithms.

Abstract

The use of denoisers for image reconstruction has shown significant potential, especially for the Plug-and-Play (PnP) framework. In PnP, a powerful denoiser is used as an implicit regularizer in proximal algorithms such as ISTA and ADMM. The focus of this work is on the convergence of PnP iterates for linear inverse problems using kernel denoisers. It was shown in prior work that the update operator in standard PnP is contractive for symmetric kernel denoisers under appropriate conditions on the denoiser and the linear forward operator. Consequently, we could establish global linear convergence of the iterates using the contraction mapping theorem. In this work, we develop a unified framework to establish global linear convergence for symmetric and nonsymmetric kernel denoisers. Additionally, we derive quantitative bounds on the contraction factor (convergence rate) for inpainting, deblurring, and superresolution. We present numerical results to validate our theoretical findings.

Paper Structure

This paper contains 20 sections, 15 theorems, 77 equations, 9 figures.

Table of Contents

  1. Introduction
  2. PnP algorithms
  3. Contributions
  4. Organization
  5. Notation
  6. Background
  7. Kernel denoiser
  8. Prior work
  9. A counterexample
  10. PnP with Kernel Denoisers
  11. Linear Convergence of PnP
  12. Contraction Factor
  13. Contraction factor for ISTA
  14. Contraction factor for ADMM
  15. Discussion
  16. ...and 5 more sections

Key Result

Proposition 1

If Assumption assumpK holds, then

Figures (9)

  • Figure 1: Deblurring results using PnP-ISTA with a symmetric DSG-NLM denoiser and a nonsymmetric NLM denoiser, and Sc-PnP-ISTA with a nonsymmetric NLM denoiser. The input image is blurred with a Gaussian kernel of size $25\times 25$ and a standard deviation of $1.6$, followed by the addition of $3\%$ white Gaussian noise. The PSNR values are (b) $22.85$ dB, (c) $29.39$ dB, (d) $28.93$ dB and (e) $28.81$ dB.
  • Figure 2: $2\times$ superresolution using PnP-ISTA and Sc-PnP-ISTA, with DSG-NLM and NLM denoisers respectively. The input is generated by applying a uniform blur with a $9 \times 9$ kernel, followed by $2\times$ downsampling and adding $3\%$ white Gaussian noise. The PSNR values are (c) $27.31$ dB, (d) $26.67$ dB, and (d) $26.49$ dB. For reference, the PSNR from bicubic interpolation is $22.80$ dB.
  • Figure 3: We compare the reconstruction quality obtained using ISTA with symmetric DSG-NLM and nonsymmetric NLM denoisers. The results are averaged over images from the Set12 dataset. For superresolution, the PSNR gain is computed with respect to the bicubic interpolation of the observed image. Overall, the symmetric denoiser DSG-NLM outperforms the nonsymmetric NLM denoiser, although it requires more computations sreehari2016plug.
  • Figure 4: We report the contraction factors of the update operators $\mathbf{P}$ and $\mathbf{P}_s$ in PnP-ISTA and Sc-PnP-ISTA for image inpainting. The contraction factors are shown for different fractions of observed pixels $\mu$ and for varying step-sizes $\gamma$. We note that the above operators are derived from the denoiser, which in turn is computed from the input image. The plots display the average results across images from Set12, with the shaded region around each plot representing the standard deviation. The results align with the predictions of Thm. \ref{['thm:boundISTA']}, confirming that the contraction factor decreases with an increase in the fraction of observed pixels.
  • Figure 5: We study the effect of the downsampling rate (for superresolution) on the contraction factor of the update operators: $\mathbf{R}$ for PnP-ADMM and $\mathbf{R}_s$ for Sc-PnP-ADMM. The bound in Thm. \ref{['thm:inp-sr-Js']} predicts a reduction in the contraction factor with lower downsampling (i.e., more samples). The results in the plots align well with this prediction.
  • ...and 4 more figures

Theorems & Definitions (31)

  • Proposition 1
  • Remark 1
  • Theorem 1
  • Theorem 2
  • Remark 2
  • Definition 1: RNP
  • Proposition 2
  • proof
  • Lemma 1
  • Proposition 3
  • ...and 21 more