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We consider the task of image reconstruction while simultaneously decomposing the reconstructed image into components with different features. A commonly used tool for this is a variational approach with an infimal convolution of appropriate functions as a regularizer. Especially for noise corrupted observations, incorporating these functionals into the classical method of Bregman iterations provides a robust method for obtaining an overall good approximation of the true image, by stopping early the iteration according to a discrepancy principle. However, crucially, the quality of the separate components depends further on the proper choice of the regularization weights associated to the infimally convoluted functionals. Here, we propose the method of Nested Bregman iterations to improve a decomposition in a structured way. This allows to transform the task of choosing the weights into the problem of stopping the iteration according to a meaningful criterion based on normalized cross-correlation. We discuss the well-definedness and the convergence behavior of the proposed method, and illustrate its strength numerically with various image decomposition tasks employing infimal convolution functionals.

The Condat-V\~u algorithm is a widely used primal-dual method for optimizing composite objectives of three functions. Several algorithms for optimizing composite objectives of two functions are special cases of Condat-V\~u, including proximal gradient descent (PGD). It is well-known that PGD exhibits suboptimal performance, and a simple adjustment to PGD can accelerate its convergence rate from $\mathcal{O}(1/T)$ to $\mathcal{O}(1/T^2)$ on convex objectives, and this accelerated rate is optimal. In this work, we show that a simple adjustment to the Condat-V\~u algorithm allows it to recover accelerated PGD (APGD) as a special case, instead of PGD. We prove that this accelerated Condat--V\~u algorithm achieves optimal convergence rates and significantly outperforms the traditional Condat-V\~u algorithm in regimes where the Condat--V\~u algorithm approximates the dynamics of PGD. We demonstrate the effectiveness of our approach in various applications in machine learning and computational imaging.

Machine learning methods offer great promise for fast and accurate detection and prognostication of COVID-19 from standard-of-care chest radiographs (CXR) and computed tomography (CT) images. Many articles have been published in 2020 describing new machine learning-based models for both of these tasks, but it is unclear which are of potential clinical utility. In this systematic review, we search EMBASE via OVID, MEDLINE via PubMed, bioRxiv, medRxiv and arXiv for published papers and preprints uploaded from January 1, 2020 to October 3, 2020 which describe new machine learning models for the diagnosis or prognosis of COVID-19 from CXR or CT images. Our search identified 2,212 studies, of which 415 were included after initial screening and, after quality screening, 61 studies were included in this systematic review. Our review finds that none of the models identified are of potential clinical use due to methodological flaws and/or underlying biases. This is a major weakness, given the urgency with which validated COVID-19 models are needed. To address this, we give many recommendations which, if followed, will solve these issues and lead to higher quality model development and well documented manuscripts., Comment: 35 pages, 3 figures, 2 tables, updated to the period 1 January 2020 - 3 October 2020

We introduce SPRING, a novel stochastic proximal alternating linearized minimization algorithm for solving a class of non-smooth and non-convex optimization problems. Large-scale imaging problems are becoming increasingly prevalent due to advances in data acquisition and computational capabilities. Motivated by the success of stochastic optimization methods, we propose a stochastic variant of proximal alternating linearized minimization (PALM) algorithm \cite{bolte2014proximal}. We provide global convergence guarantees, demonstrating that our proposed method with variance-reduced stochastic gradient estimators, such as SAGA \cite{SAGA} and SARAH \cite{sarah}, achieves state-of-the-art oracle complexities. We also demonstrate the efficacy of our algorithm via several numerical examples including sparse non-negative matrix factorization, sparse principal component analysis, and blind image deconvolution., Comment: 28 pages, 11 page appendix

Variance reduction is a crucial tool for improving the slow convergence of stochastic gradient descent. Only a few variance-reduced methods, however, have yet been shown to directly benefit from Nesterov's acceleration techniques to match the convergence rates of accelerated gradient methods. Such approaches rely on "negative momentum", a technique for further variance reduction that is generally specific to the SVRG gradient estimator. In this work, we show that negative momentum is unnecessary for acceleration and develop a universal acceleration framework that allows all popular variance-reduced methods to achieve accelerated convergence rates. The constants appearing in these rates, including their dependence on the number of functions $n$, scale with the mean-squared-error and bias of the gradient estimator. In a series of numerical experiments, we demonstrate that versions of SAGA, SVRG, SARAH, and SARGE using our framework significantly outperform non-accelerated versions and compare favourably with algorithms using negative momentum., Comment: 33 pages

We present a uniform analysis of biased stochastic gradient methods for minimizing convex, strongly convex, and non-convex composite objectives, and identify settings where bias is useful in stochastic gradient estimation. The framework we present allows us to extend proximal support to biased algorithms, including SAG and SARAH, for the first time in the convex setting. We also use our framework to develop a new algorithm, Stochastic Average Recursive GradiEnt (SARGE), that achieves the oracle complexity lower-bound for non-convex, finite-sum objectives and requires strictly fewer calls to a stochastic gradient oracle per iteration than SVRG and SARAH. We support our theoretical results with numerical experiments that demonstrate the benefits of certain biased gradient estimators., Comment: journal version, 35 pages

This paper studies tensor-based Robust Principal Component Analysis (RPCA) using atomic-norm regularization. Given the superposition of a sparse and a low-rank tensor, we present conditions under which it is possible to exactly recover the sparse and low-rank components. Our results improve on existing performance guarantees for tensor-RPCA, including those for matrix RPCA. Our guarantees also show that atomic-norm regularization provides better recovery for tensor-structured data sets than other approaches based on matricization. In addition to these performance guarantees, we study a nonconvex formulation of the tensor atomic-norm and identify a class of local minima of this nonconvex program that are globally optimal. We demonstrate the strong performance of our approach in numerical experiments, where we show that our nonconvex model reliably recovers tensors with ranks larger than all of their side lengths, significantly outperforming other algorithms that require matricization., Comment: 39 pages, 3 figures, 3 tables

We introduce a reformulation of regularized low-rank recovery models to take advantage of GPU, multiple CPU, and hybridized architectures. Low-rank recovery often involves nuclear-norm minimization through iterative thresholding of singular values. These models are slow to fit and difficult to parallelize because of their dependence on computing a singular value decomposition at each iteration. Regularized low-rank recovery models also incorporate non-smooth terms to separate structured components (e.g. sparse outliers) from the low-rank component, making these problems more difficult. Using Burer-Monteiro splitting and marginalization, we develop a smooth, non-convex formulation of regularized low-rank recovery models that can be fit with first-order solvers. We develop a computable certificate of convergence for this non-convex program, and use it to establish bounds on the suboptimality of any point. Using robust principal component analysis (RPCA) as an example, we include numerical experiments showing that this approach is an order-of-magnitude faster than existing RPCA solvers on the GPU. We also show that this acceleration allows new applications for RPCA, including real-time background subtraction and MR image analysis., Comment: 23 pages, 14 figures, pre-print

The influence of fixed temperature and fixed heat flux thermal boundary conditions on rapidly rotating convection in the plane layer geometry is investigated for the case of stress-free mechanical boundary conditions. It is shown that whereas the leading order system satisfies fixed temperature boundary conditions implicitly, a double boundary layer structure is necessary to satisfy the fixed heat flux thermal boundary conditions. The boundary layers consist of a classical Ekman layer adjacent to the solid boundaries that adjust viscous stresses to zero, and a layer in thermal wind balance just outside the Ekman layers adjusts the temperature such that the fixed heat flux thermal boundary conditions are satisfied. The influence of these boundary layers on the interior geostrophically balanced convection is shown to be asymptotically weak, however. Upon defining a simple rescaling of the thermal variables, the leading order reduced system of governing equations are therefore equivalent for both boundary conditions. These results imply that any horizontal thermal variation along the boundaries that varies on the scale of the convection has no leading order influence on the interior convection.