Related papers: Errors of regularisation under range inclusions us…
We consider Proximal Newton methods with an inexact computation of update steps. To this end, we introduce two inexactness criteria which characterize sufficient accuracy of these update step and with the aid of these investigate global…
Inexact Newton regularization methods have been proposed by Hanke and Rieder for solving nonlinear ill-posed inverse problems. Every such a method consists of two components: an outer Newton iteration and an inner scheme providing…
We obtain upper bounds for the numerical radius of a product of Hilbert space operators which improve on the existing upper bounds. We generalize the numerical radius inequalities of $n\times n$ operator matrices by using non-negative…
Many imaging problems require solving an inverse problem that is ill-conditioned or ill-posed. Imaging methods typically address this difficulty by regularising the estimation problem to make it well-posed. This often requires setting the…
Sparsity promoting regularization is an important technique for signal reconstruction and several other ill-posed problems. Theoretical investigation typically bases on the assumption that the unknown solution has a sparse representation…
Fibre Bragg Gratings have become widespread measurement devices in engineering and other fields of application. In all but a few cases, the relation between cause and effect is simplified to a proportional model. However, at its…
Under mild assumptions on the kernel, we obtain the best known error rates in a regularized learning scenario taking place in the corresponding reproducing kernel Hilbert space (RKHS). The main novelty in the analysis is a proof that one…
In this paper we revisit the discrepancy principle for Tikhonov regularization of nonlinear ill-posed problems in Hilbert spaces and provide some new and improved saturation results under less restrictive conditions, comparing with the…
We obtain bounds on estimation error rates for regularization procedures of the form \begin{equation*} \hat f \in {\rm argmin}_{f\in F}\left(\frac{1}{N}\sum_{i=1}^N\left(Y_i-f(X_i)\right)^2+\lambda \Psi(f)\right) \end{equation*} when $\Psi$…
The authors study statistical linear inverse problems in Hilbert spaces. Approximate solutions are sought within a class of linear one-parameter regularization schemes, and the parameter choice is crucial to control the root mean squared…
This paper deals with probabilistic upper bounds for the error in functional estimation defined on some interpolation and extrapolation designs, when the function to estimate is supposed to be analytic. The error pertaining to the estimate…
Regularization-based approaches for injecting constraints in Machine Learning (ML) were introduced to improve a predictive model via expert knowledge. We tackle the issue of finding the right balance between the loss (the accuracy of the…
The concept of refinement from probability elicitation is considered for proper scoring rules. Taking directions from the axioms of probability, refinement is further clarified using a Hilbert space interpretation and reformulated into the…
This article implements a simple convex approach and block techniques to obtain several new refined versions of numerical radius inequalities for Hilbert space operators. This includes comparisons among the norms of the operators, their…
Proper splittings of operators are commonly used to study the convergence of iterative processes. In order to approximate solutions of operator equations, in this article we deal with proper splittings of closed range bounded linear…
We present a method for computing nearly singular integrals that occur when single or double layer surface integrals, for harmonic potentials or Stokes flow, are evaluated at points nearby. Such values could be needed in solving an integral…
In this paper we the formulation of inverse problems as constrained minimization problems and their iterative solution by gradient or Newton type. We carry out a convergence analysis in the sense of regularization methods and discuss…
Trace norm regularization is a popular method of multitask learning. We give excess risk bounds with explicit dependence on the number of tasks, the number of examples per task and properties of the data distribution. The bounds are…
Error bounds have been studied for more than seventy years, beginning with the seminal result of Hoffman (1952) [{\it J. Res. Natl. Bur. Standards}, 49 (1952), 263--265], which establishes an upper bound for the distance from an arbitrary…
The analysis of Tikhonov regularization for nonlinear ill-posed equations with smoothness promoting penalties is an important topic in inverse problem theory. With focus on Hilbert scale models, the case of oversmoothing penalties, i.e.,…