Related papers: Variational properties of value functions
This paper discusses basic results and recent developments on variational regularization methods, as developed for inverse problems. In a typical setup we review basic properties needed to obtain a convergent regularization scheme and…
The choice of the parameter value for regularized inverse problems is critical to the results and remains a topic of interest. This article explores a criterion for selecting a good parameter value by maximizing the probability of the data,…
Various problems in computer vision and medical imaging can be cast as inverse problems. A frequent method for solving inverse problems is the variational approach, which amounts to minimizing an energy composed of a data fidelity term and…
We consider the differentiation of the value function for parametric optimization problems. Such problems are ubiquitous in Machine Learning applications such as structured support vector machines, matrix factorization and min-min or…
We establish a general principle which states that regularizing an inverse problem with a convex function yields solutions which are convex combinations of a small number of atoms. These atoms are identified with the extreme points and…
We introduce and study a mathematical framework for a broad class of regularization functionals for ill-posed inverse problems: Regularization Graphs. Regularization graphs allow to construct functionals using as building blocks linear…
We propose a new class of convex penalty functions, called \emph{variational Gram functions} (VGFs), that can promote pairwise relations, such as orthogonality, among a set of vectors in a vector space. These functions can serve as…
We investigate the computation of the gradient of the value function in parametric convex optimization problems. We derive general expression for the gradient of the value function in terms of the cost function, constraints and Lagrange…
An adaptive regularization algorithm for unconstrained nonconvex optimization is proposed that is capable of handling inexact objective-function and derivative values, and also of providing approximate minimizer of arbitrary order. In…
Singular value decomposition is the key tool in the analysis and understanding of linear regularization methods. In the last decade nonlinear variational approaches such as $\ell^1$ or total variation regularizations became quite prominent…
Variational regularization is commonly used to solve linear inverse problems, and involves augmenting a data fidelity by a regularizer. The regularizer is used to promote a priori information and is weighted by a regularization parameter.…
Much of statistics relies upon four key elements: a law of large numbers, a calculus to operationalize stochastic convergence, a central limit theorem, and a framework for constructing local approximations. These elements are…
Metric regularity is among the central concepts of nonlinear and variational analysis, constrained optimization, and their numerous applications. However, metric regularity can be elusive for some important ill-posed classes of problems…
We consider an optimization problem with positively homogeneous functions in its objective and constraint functions. Examples of such positively homogeneous functions include the absolute value function and the $p$-norm function, where $p$…
A parametrized convex function depends on a variable and a parameter, and is convex in the variable for any valid value of the parameter. Such functions can be used to specify parametrized convex optimization problems, i.e., a convex…
A regularization algorithm using inexact function values and inexact derivatives is proposed and its evaluation complexity analyzed. This algorithm is applicable to unconstrained problems and to problems with inexpensive constraints (that…
Many problems in nonlinear analysis and optimization, among them variational inequalities and minimization of convex functions, can be reduced to finding zeros (namely, roots) of set-valued operators. Hence numerous algorithms have been…
We propose an adaptive regularization scheme in a variational framework where a convex composite energy functional is optimized. We consider a number of imaging problems including denoising, segmentation and motion estimation, which are…
We deal with the convergence of the value function of an approximate control problem with uncertain dynamics to the value function of a nonlinear optimal control problem. The assumptions on the dynamics and the costs are rather general and…
Convergence rates results for variational regularization methods typically assume the regularization functional to be convex. While this assumption is natural for scalar-valued functions, it can be unnecessarily strong for vector-valued…