In this paper we propose a general framework to characterize and solve the optimization problems underlying a large class of sparsity based regularization algorithms. More precisely, we study the minimization of learning functionals that are sums of a differentiable data term and a convex non differentiable penalty. These latter penalties have recently become popular in machine learning since t...

In this paper, we consider and analyze some new projection-proximal methods for solving general variational inequalities. The modified methods converge for pseudomonotone operators which is a weaker condition than monotonicity. The proposed methods include several new and known methods as special cases. Our results can be considered as a novel and important extension of the previously known res...

The standard approach to exploiting motion elds in block-based hybrid video coding schemes is motion compensation of the current frame using the motion eld, followed by DCT coding of the residue. In this paper we argue that this separation of approximation methods (non-linear prediction followed by transform coding) is unfortunate in low bit-rate applications. In low bit-rate applications, only...

In applications like medical imaging, error correction, and sensor networks, one needs to solve large-scale linear systems that may be corrupted by a small number of, but arbitrarily large, corruptions. We consider solving such large-scale systems of linear equations $A\mathbf{x}=\mathbf{b}$ that are inconsistent due to corruptions in the measurement vector $\mathbf{b}$. With this as our motiva...

Projection pursuit was originally introduced to identify structures in multivariate data clouds (Huber, 1985). The idea of projecting data to a lowdimensional subspace can also be applied to multivariate statistical methods. The robustness of the methods can be achieved by applying robust estimators to the lower-dimensional space. Robust estimation in high dimensions can thus be avoided which u...

Abstract. This work establishes a formal derivation of local projection stabilized methods as a result of an enriched Petrov-Galerkin strategy for the Stokes problem. Both velocity and pressure finite element spaces are enhanced with solutions of residual-based local problems, and then the static condensation procedure is applied to derive new methods. The approach keeps degrees of freedom unch...

We consider the problem of constructing quantum channels, if they exist, that transform a given set of quantum states {ρ1, . . . , ρk} to another such set {ρ̂1, . . . , ρ̂k}. In other words, we must find a completely positive linear map, if it exists, that maps a given set of density matrices to another given set of density matrices, possibly of different dimension. Using the theory of completely...

We provide a unifying projection-based framework for structure-preserving interpolatory model reduction of parameterized linear dynamical systems, i.e., systems having a structured dependence on parameters that we wish to retain in the reduced-order model. The parameter dependence may be linear or nonlinear and is retained in the reduced-order model. Moreover, we are able to give conditions und...

In this paper, we propose some new double-projection methods for solving variational inequalities by using the Wiener–Hopf equations technique. It is shown that these methods converge linearly under mild conditions and include some existing projection methods as special cases. Some examples are given to illustrate the efficiency of the proposed methods. 2002 Elsevier Science Inc. All rights res...