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We introduce and study a mathematical framework for a broad class of regularization functionals for ill-posed inverse problems: Regularization Graphs. Regularization graphs allow to construct functionals using as building blocks linear…

Motivated by value function estimation in reinforcement learning, we study statistical linear inverse problems, i.e., problems where the coefficients of a linear system to be solved are observed in noise. We consider penalized estimators,…

In this paper, we propose two algorithms for solving linear inverse problems when the observations are corrupted by noise. A proper data fidelity term (log-likelihood) is introduced to reflect the statistics of the noise (e.g. Gaussian,…

Imaging inverse problems aim to recover high-dimensional signals from undersampled, noisy measurements, a fundamentally ill-posed task with infinite solutions in the null-space of the sensing operator. To resolve this ambiguity, prior…

So-called functional error estimators provide a valuable tool for reliably estimating the discretization error for a sum of two convex functions. We apply this concept to Tikhonov regularization for the solution of inverse problems for…

This paper provides a variational analysis of the unconstrained formulation of the LASSO problem, ubiquitous in statistical learning, signal processing, and inverse problems. In particular, we establish smoothness results for the optimal…

In this paper we extend a recent idea of formulating and regularizing inverse problems as minimization problems, so without using a forward operator, thus avoiding explicit evaluation of a parameter-to-state map. We do so by rephrasing…

We consider the problem of reconstructing an infinite set of sparse, finite-dimensional vectors, that share a common sparsity pattern, from incomplete measurements. This is in contrast to the work [17], where the single vector signal can be…

Statistical decision problems lie at the heart of statistical machine learning. The simplest problems are binary and multiclass classification and class probability estimation. Central to their definition is the choice of loss function,…

This paper develops a spatially resolved perturbation theory for singular vectors under high-dimensional separable noise and applies it to data-driven matrix recovery. In the asymptotic regime where the matrix dimensions are proportional…

This note addresses the question of optimally estimating a linear functional of an object acquired through linear observations corrupted by random noise, where optimality pertains to a worst-case setting tied to a symmetric, convex, and…

Convex functionals are ubiquitous in applied analysis, appearing as value functions, risk measures, super-hedging prices, and loss functionals in machine learning. In many applications, however, the functional is only observed through…

In the context of linear regression, we construct a data-driven convex loss function with respect to which empirical risk minimisation yields optimal asymptotic variance in the downstream estimation of the regression coefficients. At the…

In this paper, we study the phase retrieval problem in the situation where the vector to be recovered has an a priori structure that can encoded into a regularization term. This regularizer is intended to promote solutions conforming to…

We present a novel approach to variational volume reconstruction from sparse, noisy slice data using the Deep Ritz method. Motivated by biomedical imaging applications such as MRI-based slice-to-volume reconstruction (SVR), our approach…

In this paper, we propose a novel bounded asymmetric elastic net ($L_{baen}$) loss function and combine it with the support vector machine (SVM), resulting in the BAEN-SVM. The $L_{baen}$ is bounded and asymmetric and can degrade to the…

Designing appropriate variational regularization schemes is a crucial part of solving inverse problems, making them better-posed and guaranteeing that the solution of the associated optimization problem satisfies desirable properties.…

In the literature, necessary and sufficient conditions in terms of variational inequalities are introduced to characterize minimizers of convex set valued functions with values in a conlinear space. Similar results are proved for a weaker…

Recovering jointly sparse signals in the multiple measurement vectors (MMV) setting is a fundamental problem in machine learning, but traditional methods often require careful parameter tuning or prior knowledge of the sparsity of the…

We investigate the stochastic optimization problem of minimizing population risk, where the loss defining the risk is assumed to be weakly convex. Compositions of Lipschitz convex functions with smooth maps are the primary examples of such…