arXiv subjects

Andrea Ebner

Publications and source records attributed to Andrea Ebner.

6 recordsLinked to original sources

arXiv2026

Stochastic variance reduced extragradient methods for solving hierarchical variational inequalities

We are concerned with optimization in a broad sense through the lens of solving variational inequalities (VIs) -- a class of problems that are so general that they cover as particular cases minimization of functions, saddle-point (minimax) problems, Nash equilibrium problems, and many others. The key challenges in our problem formulation are the two-level hierarchical structure and finite-sum representation of the smooth operators in each level. For this setting, we are the first to prove convergence rates and complexity statements for variance-reduced stochastic algorithms approaching the solution of hierarchical VIs in Euclidean and Bregman setups.

math.OC↗

arXiv2024

Error Estimates for Data-driven Weakly Convex Frame-based Image Regularization

Inverse problems are fundamental in fields like medical imaging, geophysics, and computerized tomography, aiming to recover unknown quantities from observed data. However, these problems often lack stability due to noise and ill-conditioning, leading to inaccurate reconstructions. To mitigate these issues, regularization methods are employed, introducing constraints to stabilize the inversion process and achieve a meaningful solution. Recent research has shown that the application of regularizing filters to diagonal frame decompositions (DFD) yields regularization methods. These filters dampen some frame coefficients to prevent noise amplification. This paper introduces a non-linear filtered DFD method combined with a learning strategy for determining optimal non-linear filters from training data pairs. In our experiments, we applied this approach to the inversion of the Radon transform using 500 image-sinogram pairs from real CT scans. Although the learned filters were found to be strictly increasing, they did not satisfy the non-expansiveness condition required to link them with convex regularizers and prove stability and convergence in the sense of regularization methods in previous works. Inspired by this, the paper relaxes the non-expansiveness condition, resulting in weakly convex regularization. Despite this relaxation, we managed to derive stability, convergence, and convergence rates with respect to the absolute symmetric Bregman distance for the learned non-linear regularizing filters. Extensive numerical results demonstrate the effectiveness of the proposed method in achieving stable and accurate reconstructions.

math.NA↗

arXiv2023

Convergence of non-linear diagonal frame filtering for regularizing inverse problems

Inverse problems are key issues in several scientific areas, including signal processing and medical imaging. Since inverse problems typically suffer from instability with respect to data perturbations, a variety of regularization techniques have been proposed. In particular, the use of filtered diagonal frame decompositions has proven to be effective and computationally efficient. However, existing convergence analysis applies only to linear filters and a few non-linear filters such as soft thresholding. In this paper, we analyze filtered diagonal frame decompositions with general non-linear filters. In particular, our results generalize SVD-based spectral filtering from linear to non-linear filters as a special case. As a first approach, we establish a connection between non-linear diagonal frame filtering and variational regularization, allowing us to use results from variational regularization to derive the convergence of non-linear spectral filtering. In the second approach, as our main theoretical results, we relax the assumptions involved in the variational case while still deriving convergence. Furthermore, we discuss connections between non-linear filtering and plug-and-play regularization and explore potential benefits of this relationship.

math.NA↗

arXiv2022

Plug-and-Play image reconstruction is a convergent regularization method

Non-uniqueness and instability are characteristic features of image reconstruction processes. As a result, it is necessary to develop regularization methods that can be used to compute reliable approximate solutions. A regularization method provides of a family of stable reconstructions that converge to an exact solution of the noise-free problem as the noise level tends to zero. The standard regularization technique is defined by variational image reconstruction, which minimizes a data discrepancy augmented by a regularizer. The actual numerical implementation makes use of iterative methods, often involving proximal mappings of the regularizer. In recent years, Plug-and-Play image reconstruction (PnP) has been developed as a new powerful generalization of variational methods based on replacing proximal mappings by more general image denoisers. While PnP iterations yield excellent results, neither stability nor convergence in the sense of regularization has been studied so far. In this work, we extend the idea of PnP by considering families of PnP iterations, each being accompanied by its own denoiser. As our main theoretical result, we show that such PnP reconstructions lead to stable and convergent regularization methods. This shows for the first time that PnP is mathematically equally justified for robust image reconstruction as variational methods

math.NA↗

arXiv2022

Regularization of Inverse Problems by Filtered Diagonal Frame Decomposition

The characteristic feature of inverse problems is their instability with respect to data perturbations. In order to stabilize the inversion process, regularization methods have to be developed and applied. In this work we introduce and analyze the concept of filtered diagonal frame decomposition which extends the standard filtered singular value decomposition to the frame case. Frames as generalized singular system allows to better adapt to a given class of potential solutions. In this paper, we show that filtered diagonal frame decomposition yield a convergent regularization method. Moreover, we derive convergence rates under source type conditions and prove order optimality under the assumption that the considered frame is a Riesz-basis.

math.NA↗

arXiv2022

Convergence rates for the joint solution of inverse problems with compressed sensing data

Compressed sensing (CS) is a powerful tool for reducing the amount of data to be collected while maintaining high spatial resolution. Such techniques work well in practice and at the same time are supported by solid theory. Standard CS results assume measurements to be made directly on the targeted signal. In many practical applications, however, CS information can only be taken from indirect data $h_\star = W x_\star$ related to the original signal by an additional forward operator. If inverting the forward operator is ill-posed, then existing CS theory is not applicable. In this paper, we address this issue and present two joint reconstruction approaches, namely relaxed $\ell^1$ co-regularization and strict $\ell^1$ co-regularization, for CS from indirect data. As main results, we derive error estimates for recovering $x_\star$ and $h_\star$. In particular, we derive a linear convergence rate in the norm for the latter. To obtain these results, solutions are required to satisfy a source condition and the CS measurement operator is required to satisfy a restricted injectivity condition. We further show that these conditions are not only sufficient but even necessary to obtain linear convergence.

cs.IT↗