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In a recent paper by A. Chambolle et al. [Geometric properties of solutions to the total variation denoising problem. Inverse Problems 33, 2017] it was proven that if the subgradient of the total variation at the noise free data is not…
Inverse problems in physical or biological sciences often involve recovering an unknown parameter that is random. The sought-after quantity is a probability distribution of the unknown parameter, that produces data that aligns with…
We introduce Stochastic Asymptotical Regularization (SAR) methods for the uncertainty quantification of the stable approximate solution of ill-posed linear-operator equations, which are deterministic models for numerous inverse problems in…
In this paper we develop new applications of variational analysis and generalized differentiation to the following optimization problem and its specifications: given n closed subsets of a Banach space, find such a point for which the sum of…
We study generalized games defined over Banach spaces using variational analysis. To reformulate generalized games as quasi-variational inequality problems, we will first form a suitable principal operator and study some significant…
We introduce a novel family of invariant, convex, and non-quadratic functionals that we employ to derive regularized solutions of ill-posed linear inverse imaging problems. The proposed regularizers involve the Schatten norms of the Hessian…
In this paper, a methodology is investigated for signal recovery in the presence of non-Gaussian noise. In contrast with regularized minimization approaches often adopted in the literature, in our algorithm the regularization parameter is…
We consider a convex optimization problem with many linear inequality constraints. To deal with a large number of constraints, we provide a penalty reformulation of the problem, where the penalty is a variant of the one-sided Huber loss…
This paper addresses the study of novel constructions of variational analysis and generalized differentiation that are appropriate for characterizing robust stability properties of constrained set-valued mappings/multifunctions between…
In this work, a method for obtaining pixel-wise error bounds in Bayesian regularization of inverse imaging problems is introduced. The proposed method employs estimates of the posterior variance together with techniques from conformal…
Bayesian inference for inverse problems involves computing expectations under posterior distributions -- e.g., posterior means, variances, or predictive quantities -- typically via Monte Carlo (MC) estimation. When the quantity of interest…
In this paper, we deal with nonlinear ill-posed problems involving monotone operators and consider Lavrentiev's regularization method. This approach, in contrast to Tikhonov's regularization method, does not make use of the adjoint of the…
The current paper presents a novel machinery for studying non-asymptotic minimax estimation of high-dimensional matrices, which yields tight minimax rates for a large collection of loss functions in a variety of problems. Based on the…
Proximal gradient methods are a popular tool for the solution of structured, nonsmooth minimization problems. In this work, we investigate an extension of the former to general Banach spaces and provide worst-case convergence rates for,…
The new version of a posteriori choice (NVAC) of the regularization parameter in the classical Tikhonov regularization method is considered. Lemmas and theorems on the error and the asymptotic convergence rate of the regularized solution…
We consider the inverse problem of reconstructing inhomogeneities by performing a finite number of scattering measurements of acoustic type in the time-harmonic setting. We set up the reconstruction as a fully discrete variational problem…
In this paper we consider a stochastic heavy-ball method for solving linear ill-posed inverse problems. With suitable choices of the step-sizes and the momentum coefficients, we establish the regularization property of the method under {\it…
Landweber-type methods are prominent for solving ill-posed inverse problems in Banach spaces and their convergence has been well-understood. However, how to derive their convergence rates remains a challenging open question. In this paper,…
The article addresses a long-standing open problem on the justification of using variational Bayes methods for parameter estimation. We provide general conditions for obtaining optimal risk bounds for point estimates acquired from…
We propose a scalable variational Bayes method for statistical inference for a single or low-dimensional subset of the coordinates of a high-dimensional parameter in sparse linear regression. Our approach relies on assigning a mean-field…