Dirk Lorenz

Sebastian Banert, Christoph Brauer, Dirk Lorenz et al.

This article addresses the challenge of learning effective regularizers for linear inverse problems. We analyze and compare several types of learned variational regularization against the theoretical benchmark of the optimal affine reconstruction, i.e. the best possible affine linear map for minimizing the mean squared error. It is known that this optimal reconstruction can be achieved using Tikhonov regularization, but this requires precise knowledge of the noise covariance to properly weight the data fidelity term. However, in many practical applications, noise statistics are unknown. We therefore investigate the performance of regularization methods learned without access to this noise information, focusing on Tikhonov, Lavrentiev, and quadratic regularization. Our theoretical analysis and numerical experiments demonstrate that for non-white noise, a performance gap emerges between these methods and the optimal affine reconstruction. Furthermore, we show that these different types of regularization yield distinct results, highlighting that the choice of regularizer structure is critical when the noise model is not explicitly learned. Our findings underscore the significant value of accurately modeling or co-learning noise statistics in data-driven regularization.

Christoph Brauer, Dirk Lorenz

In this work, we propose a deep neural network architecture motivated by primal-dual splitting methods from convex optimization. We show theoretically that there exists a close relation between the derived architecture and residual networks, and further investigate this connection in numerical experiments. Moreover, we demonstrate how our approach can be used to unroll optimization algorithms for certain problems with hard constraints. Using the example of speech dequantization, we show that our method can outperform classical splitting methods when both are applied to the same task.

Christoph Brauer, Christian Clason, Dirk Lorenz et al.

We consider the entropic regularization of discretized optimal transport and propose to solve its optimality conditions via a logarithmic Newton iteration. We show a quadratic convergence rate and validate numerically that the method compares favorably with the more commonly used Sinkhorn--Knopp algorithm for small regularization strength. We further investigate numerically the robustness of the proposed method with respect to parameters such as the mesh size of the discretization.

Bangti Jin, Dirk Lorenz

In this paper, we are interested in heuristic parameter choice rules for general convex variational regularization which are based on error estimates. Two such rules are derived and generalize those from quadratic regularization, namely the Hanke-Raus rule and quasi-optimality criterion. A posteriori error estimates are shown for the Hanke-Raus rule, and convergence for both rules is also discussed. Numerical results for both rules are presented to illustrate their applicability.

Bangti Jin, Dirk Lorenz, Stefan Schiffler

This paper investigates theoretical properties and efficient numerical algorithms for the so-called elastic-net regularization originating from statistics, which enforces simultaneously l^1 and l^2 regularization. The stability of the minimizer and its consistency are studied, and convergence rates for both a priori and a posteriori parameter choice rules are established. Two iterative numerical algorithms of active set type are proposed, and their convergence properties are discussed. Numerical results are presented to illustrate the features of the functional and algorithms.