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We consider the problem of sparse atomic optimization, where the notion of "sparsity" is generalized to meaning some linear combination of few atoms. The definition of atomic set is very broad; popular examples include the standard basis,…
This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible. Numerous renowned algorithms for tackling the compressed sensing problem…
Motivated by the extensive application of approximate gradients in machine learning and optimization, we investigate inexact subgradient methods subject to persistent additive errors. Within a nonconvex semialgebraic framework, assuming…
We propose an efficient algorithm for sparse signal reconstruction problems. The proposed algorithm is an augmented Lagrangian method based on the dual sparse reconstruction problem. It is efficient when the number of unknown variables is…
We present new results for the Frank-Wolfe method (also known as the conditional gradient method). We derive computational guarantees for arbitrary step-size sequences, which are then applied to various step-size rules, including simple…
We propose a novel Stochastic Frank-Wolfe (a.k.a. conditional gradient) algorithm for constrained smooth finite-sum minimization with a generalized linear prediction/structure. This class of problems includes empirical risk minimization…
This study develops an algorithm for distributed computing of linear programming problems of huge-scales. Global consensus with single common variable, multiblocks, and augmented Lagrangian are adopted. The consensus is used to partition…
In this paper we propose and analyze inexact and stochastic versions of the CGALP algorithm developed in the authors' previous paper, which we denote ICGALP, that allows for errors in the computation of several important quantities. In…
The conjugate gradient method is a widely used algorithm for the numerical solution of a system of linear equations. It is particularly attractive because it allows one to take advantage of sparse matrices and produces (in case of infinite…
This paper focuses on the problem of \emph{constrained} \emph{stochastic} optimization. A zeroth order Frank-Wolfe algorithm is proposed, which in addition to the projection-free nature of the vanilla Frank-Wolfe algorithm makes it gradient…
Motivated by variational inference methods, we propose a zeroth-order algorithm for solving optimization problems in the space of Gaussian probability measures. The algorithm is based on an interacting system of Gaussian particles that…
This paper investigates the problem of recovering missing samples using methods based on sparse representation adapted especially for image signals. Instead of $l_2$-norm or Mean Square Error (MSE), a new perceptual quality measure is used…
The celebrated sparse representation model has led to remarkable results in various signal processing tasks in the last decade. However, despite its initial purpose of serving as a global prior for entire signals, it has been commonly used…
Min-max formulations have attracted great attention in the ML community due to the rise of deep generative models and adversarial methods, while understanding the dynamics of gradient algorithms for solving such formulations has remained a…
We develop a methodology for closing duality gap and guaranteeing strong duality in infinite convex optimization. Specifically, we examine two new Lagrangian-type dual formulations involving infinitely many dual variables and infinite sums…
Regularization of ill-posed linear inverse problems via $\ell_1$ penalization has been proposed for cases where the solution is known to be (almost) sparse. One way to obtain the minimizer of such an $\ell_1$ penalized functional is via an…
The recently introduced Gradient Methods with Memory use a subset of the past oracle information to create an accurate model of the objective function that enables them to surpass the Gradient Method in practical performance. The model…
We introduce a framework, which we denote as the augmented estimate sequence, for deriving fast algorithms with provable convergence guarantees. We use this framework to construct a new first-order scheme, the Accelerated Composite Gradient…
Sparsity constrained minimization captures a wide spectrum of applications in both machine learning and signal processing. This class of problems is difficult to solve since it is NP-hard and existing solutions are primarily based on…
This paper presents a comprehensive analysis of the well-known extragradient (EG) method for solving both equations and inclusions. First, we unify and generalize EG for [non]linear equations to a wider class of algorithms, encompassing…