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1. Block Coordinate Regularization by Denoising

   Sun, Yu ; Liu, Jiaming ; Kamilov, Ulugbek S ( January 2019 , Advances in neural information processing systems)

   We consider the problem of estimating a vector from its noisy measurements using a prior specified only through a denoising function. Recent work on plug- and-play priors (PnP) and regularization-by-denoising (RED) has shown the state- of-the-art performance of estimators under such priors in a range of imaging tasks. In this work, we develop a new block coordinate RED algorithm that decomposes a large-scale estimation problem into a sequence of updates over a small subset of the unknown variables. We theoretically analyze the convergence of the algorithm and discuss its relationship to the traditional proximal optimization. Our analysis complements and extends recent theoretical results for RED-based estimation methods. We numerically validate our method using several denoiser priors, including those based on convolutional neural network (CNN) denoisers. 

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2. Async-RED: A Provably Convergent Asynchronous Block Parallel Stochastic Method using Deep Denoising Priors

   Sun, Yu ; Liu, Jiaming ; Sun, Yiran ; Wohlberg, Brendt ; Kamilov, Ulugbek S. ( May 2021 , International Conference on Learning Representations)

   null (Ed.)

   Regularization by denoising (RED) is a recently developed framework for solving inverse problems by integrating advanced denoisers as image priors. Recent work has shown its state-of-the-art performance when combined with pre-trained deep denoisers. However, current RED algorithms are inadequate for parallel processing on multicore systems. We address this issue by proposing a new asynchronous RED (ASYNC-RED) algorithm that enables asynchronous parallel processing of data, making it significantly faster than its serial counterparts for large-scale inverse problems. The computational complexity of ASYNC-RED is further reduced by using a random subset of measurements at every iteration. We present complete theoretical analysis of the algorithm by establishing its convergence under explicit assumptions on the data-fidelity and the denoiser. We validate ASYNC-RED on image recovery using pre-trained deep denoisers as priors. 

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3. Monotonically Convergent Regularization by Denoising

   https://doi.org/10.1109/icip46576.2022.9897639  

   Hu, Yuyang ; Liu, Jiaming ; Xu, Xiaojian ; Kamilov, Ulugbek S. ( October 2022 , IEEE International Conference on Image Processing (ICIP))

   Regularization by denoising (RED) is a widely-used framework for solving inverse problems by leveraging image de-noisers as image priors. Recent work has reported the state-of-the-art performance of RED in a number of imaging applications using pre-trained deep neural nets as denoisers. Despite the recent progress, the stable convergence of RED algorithms remains an open problem. The existing RED theory only guarantees stability for convex data-fidelity terms and nonexpansive denoisers. This work addresses this issue by developing a new monotone RED (MRED) algorithm, whose convergence does not require nonexpansiveness of the deep denoising prior. Simulations on image deblurring and compressive sensing recovery from random matrices show the stability of MRED even when the traditional RED diverges. 

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4. Online Deep Equilibrium Learning for Regularization by Denoising

   Liu, Jiaming ; Xu, Xiaojian ; Gan, Weijie ; Shoushtari, Shirin ; Kamilov, Ulugbek S. ( January 2022 , Advances in neural information processing systems)

   Plug-and-Play Priors (PnP) and Regularization by Denoising (RED) are widely- used frameworks for solving imaging inverse problems by computing fixed-points of operators combining physical measurement models and learned image priors. While traditional PnP/RED formulations have focused on priors specified using image denoisers, there is a growing interest in learning PnP/RED priors that are end-to-end optimal. The recent Deep Equilibrium Models (DEQ) framework has enabled memory-efficient end-to-end learning of PnP/RED priors by implicitly differentiating through the fixed-point equations without storing intermediate activation values. However, the dependence of the computational/memory complexity of the measurement models in PnP/RED on the total number of measurements leaves DEQ impractical for many imaging applications. We propose ODER as a new strategy for improving the efficiency of DEQ through stochastic approximations of the measurement models. We theoretically analyze ODER giving insights into its ability to approximate the traditional DEQ approach for solving inverse problems. Our numerical results suggest the potential improvements in training/testing complexity due to ODER on three distinct imaging applications. 

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5. Recovery Analysis for Plug-and-Play Priors using the Restricted Eigenvalue Condition

   Liu, Jiaming ; Asif, M. Salman ; Wohlberg, Brendt ; Kamilov, Ulugbek ( December 2021 , Conference on Neural Information Processing Systems (NeurIPS))

   The plug-and-play priors (PnP) and regularization by denoising (RED) methods have become widely used for solving inverse problems by leveraging pre-trained deep denoisers as image priors. While the empirical imaging performance and the theoretical convergence properties of these algorithms have been widely investigated, their recovery properties have not previously been theoretically analyzed. We address this gap by showing how to establish theoretical recovery guarantees for PnP/RED by assuming that the solution of these methods lies near the fixed-points of a deep neural network. We also present numerical results comparing the recovery performance of PnP/RED in compressive sensing against that of recent compressive sensing algorithms based on generative models. Our numerical results suggest that PnP with a pre-trained artifact removal network provides significantly better results compared to the existing state-of-the-art methods. 

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