Related papers: Inverse learning in Hilbert scales
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…
We propose a new approach to linear ill-posed inverse problems. Our algorithm alternates between enforcing two constraints: the measurements and the statistical correlation structure in some transformed space. We use a non-linear multiscale…
Iterative regularization is a classic idea in regularization theory, that has recently become popular in machine learning. On the one hand, it allows to design efficient algorithms controlling at the same time numerical and statistical…
We consider an inverse problem of recovering a potential associated to a semi-linear wave equation with a quadratic nonlinearity in $1 + 1$ dimensions. We develop a numerical scheme to determine the potential from a noisy…
We study weighted Tikhonov regularization for large-scale linear discrete ill-posed problems with random noise. Under a polynomial upper-bound assumption on the generalized eigenvalues of the discrete forward operator, we derive stochastic…
We analyze gradient descent with randomly weighted data points in a linear regression model, under a generic weighting distribution. This includes various forms of stochastic gradient descent, importance sampling, but also extends to…
Operator learning offers a robust framework for approximating mappings between infinite-dimensional function spaces. It has also become a powerful tool for solving inverse problems in the computational sciences. This chapter surveys…
In many statistical linear inverse problems, one needs to recover classes of similar curves from their noisy images under an operator that does not have a bounded inverse. Problems of this kind appear in many areas of application.…
We consider the problem of positioning a cloud of points in the Euclidean space $\mathbb{R}^d$, using noisy measurements of a subset of pairwise distances. This task has applications in various areas, such as sensor network localization and…
The problem of object restoration in the case of spatially incoherent illumination is considered. A regularized solution to the inverse problem is obtained through a probabilistic approach, and a numerical algorithm based on the statistical…
Solving inverse problems with iterative algorithms is popular, especially for large data. Due to time constraints, the number of possible iterations is usually limited, potentially affecting the achievable accuracy. Given an error one is…
We discuss stability for a class of learning algorithms with respect to noisy labels. The algorithms we consider are for regression, and they involve the minimization of regularized risk functionals, such as L(f) := 1/N sum_i…
In the problem of learning mixtures of linear regressions, the goal is to learn a collection of signal vectors from a sequence of (possibly noisy) linear measurements, where each measurement is evaluated on an unknown signal drawn uniformly…
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…
Inverse problems are often ill-posed, with solutions that depend sensitively on data. In any numerical approach to the solution of such problems, regularization of some form is needed to counteract the resulting instability. This paper is…
Recovering a function or high-dimensional parameter vector from indirect measurements is a central task in various scientific areas. Several methods for solving such inverse problems are well developed and well understood. Recently, novel…
Inverse optimization is a powerful paradigm for learning preferences and restrictions that explain the behavior of a decision maker, based on a set of external signal and the corresponding decision pairs. However, most inverse optimization…
This paper addresses the scalar regression problem through a novel solution to exactly optimize the Huber loss in a general semi-supervised setting, which combines multi-view learning and manifold regularization. We propose a principled…
Statistical inverse learning theory, a field that lies at the intersection of inverse problems and statistical learning, has lately gained more and more attention. In an effort to steer this interplay more towards the variational…
This paper studies a Nystr\"om type subsampling approach to large kernel learning methods in the misspecified case, where the target function is not assumed to belong to the reproducing kernel Hilbert space generated by the underlying…