ASYNC-RED: A PROVABLY CONVERGENT ASYN-CHRONOUS BLOCK PARALLEL STOCHASTIC METHOD USING DEEP DENOISING PRIORS

Abstract

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.

1. INTRODUCTION

Imaging inverse problems seek to recover an unknown image x 2 R n from its noisy measurements y 2 R m . Such problems arise in many fields, ranging from low-level computer vision to biomedical imaging. Since many imaging inverse problems are ill-posed, it is common to regularize the solution by using prior information on the unknown image. Widely-adopted image priors include total variation, low-rank penalties, and transform-domain sparsity (Rudin et al., 1992; Figueiredo & Nowak, 2001; 2003; Hu et al., 2012; Elad & Aharon, 2006) . There has been considerable recent interest in plug-and-play priors (PnP) (Venkatakrishnan et al., 2013; Sreehari et al., 2016) and regularization by denoising (RED) (Romano et al., 2017) , as frameworks for exploiting image denoisers as priors for image recovery. The popularity of deep learning has led to a wide adoption of deep denoisers within PnP/RED, leading to their state-of-the-art performance in a variety of applications, including image restoration (Mataev et al., 2019) , phase retrieval (Metzler et al., 2018) , and tomographic imaging (Wu et al., 2020) . Their empirical success has also prompted a follow-up theoretical work clarifying the existence of explicit regularizers (Reehorst & Schniter, 2019) , providing new interpretations based on fixed-point projections (Cohen et al., 2020) , and analyzing their coordinate/online variants (Sun et al., 2019a; Wu et al., 2020) . Nonetheless, current PnP/RED algorithms are inherently serial. As illustrated in Fig. 1 , this makes them suboptimal on multicore systems that are often required for processing large-scale datasets (Recht et al., 2011) , such as those involving biomedical (Ong et al., 2020) and astronomical images (Akiyama et al., 2019) We address this gap by proposing a novel asynchronous RED (ASYNC-RED) algorithm. The algorithm decomposes the inference problem into a sequence of partial (block-coordinate) updates on x executed asynchronously in parallel over a multicore system. ASYNC-RED leads to a more efficient usage of available cores by avoiding synchronization of partial updates. ASYNC-RED is also scalable in terms of the number of measurements, since it processes only a small random subset of y at every iteration. We present two new theoretical results on the convergence of ASYNC-RED based on a unified set of explicit assumptions on the data-fidelity and the denoiser. Specifically, we establish its fixed-point convergence in the batch setting and extend this analysis to the randomized minibatch scenario. Our results extend recent work on serial block-coordinate RED (BC-RED) (Sun et al., 

