On the Proof of Fixed-Point Convergence for Plug-and-Play ADMM

In most state-of-the-art image restoration methods, the sum of a data-fidelity and a regularization term is optimized using an iterative algorithm such as ADMM (alternating direction method of multipliers). In recent years, the possibility of using denoisers for regularization has been explored in s...

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Bibliographic Details
Published in:IEEE signal processing letters 2019-12, Vol.26 (12), p.1817-1821
Main Authors: Gavaskar, Ruturaj Girish, Chaudhury, Kunal Narayan
Format: Article
Language:English
Subjects:
ADMM
Algorithms
conver-gence analysis
Convergence
Convex functions
Image restoration
Iterative algorithms
Iterative methods
Noise reduction
Optimization
plug-and-play
Regularization
Signal processing algorithms
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Summary:In most state-of-the-art image restoration methods, the sum of a data-fidelity and a regularization term is optimized using an iterative algorithm such as ADMM (alternating direction method of multipliers). In recent years, the possibility of using denoisers for regularization has been explored in several works. A popular approach is to formally replace the proximal operator within the ADMM framework with some powerful denoiser. However, since most state-of-the-art denoisers cannot be posed as a proximal operator, one cannot guarantee the convergence of these so-called plug-and-play (PnP) algorithms. In fact, the theoretical convergence of PnP algorithms is an active research topic. In this letter, we consider the result of Chan et al. (IEEE TCI, 2017), where fixed-point convergence of an ADMM-based PnP algorithm was established for a class of denoisers. We argue that the original proof is incomplete, since convergence is not analyzed for one of the three possible cases outlined in the letter. Moreover, we explain why the argument for the other cases does not apply in this case. We give a different analysis to fill this gap, which firmly establishes the original convergence theorem.
ISSN:1070-9908
1558-2361
DOI:10.1109/LSP.2019.2950611