Related papers: Convergence rates for regularization functionals w…

The Augmented Lagrangian Method as an approach for regularizing inverse problems received much attention recently, e.g. under the name Bregman iteration in imaging. This work shows convergence (rates) for this method when Morozov's…

We study the application of the Augmented Lagrangian Method to the solution of linear ill-posed problems. Previously, linear convergence rates with respect to the Bregman distance have been derived under the classical assumption of a…

In this paper we introduce two conceptual algorithms for minimising abstract convex functions. Both algorithms rely on solving a proximal-type subproblem with an abstract Bregman distance based proximal term. We prove their convergence when…

We consider a statistical inverse learning problem, where the task is to estimate a function $f$ based on noisy point evaluations of $Af$, where $A$ is a linear operator. The function $Af$ is evaluated at i.i.d. random design points $u_n$,…

We propose a novel framework for the regularised inversion of deep neural networks. The framework is based on the authors' recent work on training feed-forward neural networks without the differentiation of activation functions. The…

We consider the problem of optimizing the sum of a smooth convex function and a non-smooth convex function using proximal-gradient methods, where an error is present in the calculation of the gradient of the smooth term or in the proximity…

Motivated by a constrained minimization problem, it is studied the gradient flows with respect to Hessian Riemannian metrics induced by convex functions of Legendre type. The first result characterizes Hessian Riemannian structures on…

It is now well understood that convex programming can be used to estimate the frequency components of a spectrally sparse signal from $2m+1$ uniform temporal measurements. It is conjectured that a phase transition on the success of the…

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 consider a class of infinite-dimensional optimization problems in which a distributed vector-valued variable should pointwise almost everywhere take values from a given finite set $\mathcal{M}\subset\mathbb{R}^m$. Such hybrid…

Statistical inverse learning theory, a field that lies at the intersection of inverse problems and statistical learning, has lately gained more and more attention. In an effort to steer this interplay more towards the variational…

Variational sparsity regularization based on $\ell^1$-norms and other nonlinear functionals has gained enormous attention recently, both with respect to its applications and its mathematical analysis. A focus in regularization theory has…

All imaging modalities such as computed tomography (CT), emission tomography and magnetic resonance imaging (MRI) require a reconstruction approach to produce an image. A common image processing task for applications that utilise those…

Deep metric learning techniques have been used for visual representation in various supervised and unsupervised learning tasks through learning embeddings of samples with deep networks. However, classic approaches, which employ a fixed…

We study variational regularization methods in a general framework, more precisely those methods that use a discrepancy and a regularization functional. While several sets of sufficient conditions are known to obtain a regularization…

Bregman divergences generalize measures such as the squared Euclidean distance and the KL divergence, and arise throughout many areas of machine learning. In this paper, we focus on the problem of approximating an arbitrary Bregman…

This work addresses the problem of error concealment in video transmission systems over noisy channels employing Bregman divergences along with regularization. Error concealment intends to improve the effects of disturbances at the…

This work proposes a novel convex-non-convex formulation of the image segmentation and the image completion problems. The proposed approach is based on the minimization of a functional involving two distinct regularization terms: one…

Many descent algorithms for multiobjective optimization have been developed in the last two decades. Tanabe et al. (Comput Optim Appl 72(2):339--361, 2019) proposed a proximal gradient method for multiobjective optimization, which can solve…

In the context of convex variational regularization it is a known result that, under suitable differentiability assumptions, source conditions in the form of variational inequalities imply range conditions, while the converse implication…