In this paper, we clarify the relations between the existing sets of regularity conditions for convergence rates of nonparametric indirect regression (NPIR) and nonparametric instrumental variables (NPIV) regression models. We establish minimax risk lower bounds in mean integrated squared error loss for the NPIR and the NPIV models under two basic regularity conditions that allow for both mildl...

We propose a partially learned approach for the solution of ill posed inverse problems with not necessarily linear forward operators. The method builds on ideas from classical regularization theory and recent advances in deep learning to perform learning while making use of prior information about the inverse problem encoded in the forward operator, noise model and a regularizing functional. Th...

We consider linear ill-posed inverse problems y = Ax, in which we want to infer many count parameters x from few count observations y, where the matrix A is binary and has some unimodularity property. Such problems are typical in applications such as contingency table analysis and network tomography (on which we present testing results). These properties of A have a geometrical implication for ...

In this paper, we study the regularizing properties of the conditional stability estimates in ill-posed problems. First, we analyze how conditional stability estimates occur, and which properties the corresponding index functions must obey. In addition, we adapt the convergence analysis for the Tikhonov regularization in Banach spaces where the difference between the approximated solution and t...

We present new theoretical results which have implications in answering one of the fundamental questions in computer vision: recognition of surfaces and surface shapes. We state the conditions under which: (i) a surface can be recovered, uniquely, from the tangent plane map, in particular from the Gauss map; (ii) a surface shape can be recovered from the metric and the deforming forces. In case...

Tikhonov regularization is one of the most popular methods for solving linear systems of equations or linear least-squares problems with a severely ill-conditioned matrix A. This method replaces the given problem by a penalized least-squares problem. The present paper discusses measuring the residual error (discrepancy) in Tikhonov regularization with a seminorm that uses a fractional power of ...

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We prove Hölder-continuous dependence results for the difference between solutions of certain ill-posed and approximate well-posed problems in both Hilbert and Banach spaces. We use operator-theoretic methods, including C-semigroups, to treat the abstract Cauchy problem du dt = Au, u(0) = χ, 0 ≤ t < T, where the operator −A is the infinitesimal generator of a holomorphic semigroup.

In this article we design and analyze multilevel preconditioners for linear systems arising from regularized inverse problems. Using a scaleindependent distance function that measures spectral equivalence of operators, it is shown that these preconditioners approximate the inverse of the operator to optimal order with respect to the spatial discretization parameter h. As a consequence, the numb...

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