Meira Iske

Meira Iske, Carola-Bibiane Schönlieb

Many normalizing flow architectures impose regularity constraints, yet their distributional approximation properties are not fully characterized. We study the expressivity of bi-Lipschitz normalizing flows through the lens of score-based diffusion models. For the probability flow ODE of a variance-preserving diffusion, Lipschitz regularity of the score induces a flow of bi-Lipschitz diffeomorphic transport maps. This ODE bridge allows us to analyze the distributional approximation power of bi-Lipschitz normalizing flows and, conversely, derive deterministic convergence guarantees for diffusion-based transport. Our key idea is to use the probability flow ODE to link regularity of the score to regularity of the induced transport maps. We verify score regularity for broad target densities, including compactly supported densities, Gaussian convolutions of compactly supported measures and finite Gaussian mixtures. We obtain a universal distributional approximation result: Gaussian pullbacks induced by bi-Lipschitz variance-preserving transport maps are $L^1$-dense among all probability densities. For Gaussian convolution targets, we further obtain convergence in Kullback-Leibler divergence without early stopping.

Meira Iske, Hannes Albers, Tobias Knopp et al.

Magnetic Particle Imaging (MPI) is an emerging imaging modality based on the magnetic response of superparamagnetic iron oxide nanoparticles to achieve high-resolution and real-time imaging without harmful radiation. One key challenge in the MPI image reconstruction task arises from its underlying noise model, which does not fulfill the implicit Gaussian assumptions that are made when applying traditional reconstruction approaches. To address this challenge, we introduce the Learned Discrepancy Approach, a novel learning-based reconstruction method for inverse problems that includes a learned discrepancy function. It enhances traditional techniques by incorporating an invertible neural network to explicitly model problem-specific noise distributions. This approach does not rely on implicit Gaussian noise assumptions, making it especially suited to handle the sophisticated noise model in MPI and also applicable to other inverse problems. To further advance MPI reconstruction techniques, we introduce the MPI-MNIST dataset - a large collection of simulated MPI measurements derived from the MNIST dataset of handwritten digits. The dataset includes noise-perturbed measurements generated from state-of-the-art model-based system matrices and measurements of a preclinical MPI scanner device. This provides a realistic and flexible environment for algorithm testing. Validated against the MPI-MNIST dataset, our method demonstrates significant improvements in reconstruction quality in terms of structural similarity when compared to classical reconstruction techniques.

Arne Behrens, Meira Iske, Ming Jiang et al.

This paper presents an error analysis of classical and learned Tikhonov regularization schemes for inverse problems. We first demonstrate, both theoretically and numerically, that using a fixed regularization parameter across varying noise levels-which is a common miss-specification in practice-has only a mild impact on the reconstruction error. As a special case, we then investigate scenarios where the true data resides in an unknown finite-dimensional subspace. Here, our results lead to an empirically supported strategy for estimating the unknown dimension based on numerical experiments. Finally, we examine the approach that motivated this study: a method where a sparsity-promoting term is learned from denoising tasks and subsequently applied to general inverse problems via a simple heuristic parameter selection. The corresponding error analysis is initially developed using classical concepts and subsequently refined through a more detailed investigation of the discretized setting.