相关论文: Shearlet-based regularization in statistical inver…

This paper addresses the regularization by sparsity constraints by means of weighted $\ell^p$ penalties for $0\leq p\leq 2$. For $1\leq p\leq 2$ special attention is payed to convergence rates in norm and to source conditions. As main…

Recent advances in machine learning have inspired a surge of research into reconstructing specific quantities of interest from measurements that comply with certain physical laws. These efforts focus on inverse problems that are governed by…

Recently, the stochastic asymptotical regularization (SAR) has been developed in (\emph{Inverse Problems}, 39: 015007, 2023) for the uncertainty quantification of the stable approximate solution of linear ill-posed inverse problems. In this…

Sparse learning is a very important tool for mining useful information and patterns from high dimensional data. Non-convex non-smooth regularized learning problems play essential roles in sparse learning, and have drawn extensive attentions…

To reduce the x-ray dose in computerized tomography (CT), many constrained optimization approaches have been proposed aiming at minimizing a regularizing function that measures lack of consistency with some prior knowledge about the object…

This article delves into the study of the theory of regularized learning in Banach spaces for linear-functional data. It encompasses discussions on representer theorems, pseudo-approximation theorems, and convergence theorems. Regularized…

In this paper we consider variational regularization methods for inverse problems with large noise that is in general unbounded in the image space of the forward operator. We introduce a Banach space setting that allows to define a…

We study problem-dependent rates, i.e., generalization errors that scale near-optimally with the variance, the effective loss, or the gradient norms evaluated at the "best hypothesis." We introduce a principled framework dubbed "uniform…

Tikhonov regularization involves minimizing the combination of a data discrepancy term and a regularizing term, and is the standard approach for solving inverse problems. The use of non-convex regularizers, such as those defined by trained…

Inverse problems are inherently ill-posed and therefore require regularization techniques to achieve a stable solution. While traditional variational methods have well-established theoretical foundations, recent advances in machine learning…

We present a new approach to convergence rate results for variational regularization. Avoiding Bregman distances and using image space approximation rates as source conditions we prove a nearly minimax theorem showing that the modulus of…

Modern statistical applications often involve minimizing an objective function that may be nonsmooth and/or nonconvex. This paper focuses on a broad Bregman-surrogate algorithm framework including the local linear approximation, mirror…

Selecting the best regularization parameter in inverse problems is a classical and yet challenging problem. Recently, data-driven approaches have become popular to tackle this challenge. These approaches are appealing since they do require…

Solving inverse problems \(Ax = y\) is central to a variety of practically important fields such as medical imaging, remote sensing, and non-destructive testing. The most successful and theoretically best-understood method is convex…

Although the \emph{residual method}, or \emph{constrained regularization}, is frequently used in applications, a detailed study of its properties is still missing. This sharply contrasts the progress of the theory of Tikhonov…

This paper proposes a unified framework for the investigation of constrained learning theory in reflexive Banach spaces of features via regularized empirical risk minimization. The focus is placed on Tikhonov-like regularization with…

We consider Tikhonov-type variational regularization of ill-posed linear operator equations in Banach spaces with general convex penalty functionals. Upper bounds for certain error measures expressing the distance between exact and…

The goal of this paper is to further develop an approach to inverse problems with imperfect forward operators that is based on partially ordered spaces. Studying the dual problem yields useful insights into the convergence of the…

Inverse problems arise in a wide spectrum of applications in fields ranging from engineering to scientific computation. Connected with the rise of interest in inverse problems is the development and analysis of regularization methods, such…

This work is concerned with linear inverse problems where a distributed parameter is known a priori to only take on values from a given discrete set. This property can be promoted in Tikhonov regularization with the aid of a suitable convex…