Related papers: A Machine Learning Approach to Optimal Tikhonov Re…
Many key problems in machine learning and data science are routinely modeled as optimization problems and solved via optimization algorithms. With the increase of the volume of data and the size and complexity of the statistical models used…
The linear functional strategy for the regularization of inverse problems is considered. For selecting the regularization parameter therein, we propose the heuristic quasi-optimality principle and some modifications including the smoothness…
In this chapter we provide a theoretically founded investigation of state-of-the-art learning approaches for inverse problems from the point of view of spectral reconstruction operators. We give an extended definition of regularization…
Many inverse problems include nuisance parameters which, while not of direct interest, are required to recover primary parameters. Structure present in these problems allows efficient optimization strategies - a well known example is…
We present a novel approach to nonlinear constrained Tikhonov regularization from the viewpoint of optimization theory. A second-order sufficient optimality condition is suggested as a nonlinearity condition to handle the nonlinearity of…
We study inverse problems F(f) = g with perturbed right hand side g^{obs} corrupted by so-called impulsive noise, i.e. noise which is concentrated on a small subset of the domain of definition of g. It is well known that Tikhonov-type…
In this work, we investigate various approaches that use learning from training data to solve inverse problems, following a bi-level learning approach. We consider a general framework for optimal inversion design, where training data can be…
We consider the estimation of the regularization parameter for the simultaneous deblurring of multiple noisy images via Tikhonov regularization. We approach the problem in three ways. We first reduce the problem to a single-image deblurring…
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…
In this article we study the problem of recovering the unknown solution of a linear ill-posed problem, via iterative regularization methods. We review the problem of projection-regularization from a statistical point of view. A basic…
An ill-posed inverse problem of autoconvolution type is investigated. This inverse problem occurs in nonlinear optics in the context of ultrashort laser pulse characterization. The novelty of the mathematical model consists in a physically…
The paper considers the problem of performing a task defined on a model parameter that is only observed indirectly through noisy data in an ill-posed inverse problem. A key aspect is to formalize the steps of reconstruction and task as…
Statistical inverse learning aims at recovering an unknown function $f$ from randomly scattered and possibly noisy point evaluations of another function $g$, connected to $f$ via an ill-posed mathematical model. In this paper we blend…
The solution of inverse problems is crucial in various fields such as medicine, biology, and engineering, where one seeks to find a solution from noisy observations. These problems often exhibit non-uniqueness and ill-posedness, resulting…
This paper presents a detailed theoretical analysis of the three stochastic approximation proximal gradient algorithms proposed in our companion paper [49] to set regularization parameters by marginal maximum likelihood estimation. We prove…
Learned denoisers play a fundamental role in various signal generation (e.g., diffusion models) and reconstruction (e.g., compressed sensing) architectures, whose success derives from their ability to leverage low-dimensional structure in…
Variational regularization of ill-posed inverse problems is based on minimizing the sum of a data fidelity term and a regularization term. The balance between them is tuned using a positive regularization parameter, whose automatic choice…
We study the problem of learning the objective functions or constraints of a multiobjective decision making model, based on a set of sequentially arrived decisions. In particular, these decisions might not be exact and possibly carry…
This paper deals with the problem of finding suboptimal values of an unknown function on the basis of measured data corrupted by bounded noise. As a prior, we assume that the unknown function is parameterized in terms of a number of basis…
Inverse problems arise in a number of domains such as medical imaging, remote sensing, and many more, relying on the use of advanced signal and image processing approaches -- such as sparsity-driven techniques -- to determine their…