This thesis develops an uncertainty quantification (UQ) method for high-dimensional regression. More concretely, we construct provably valid confidence intervals for solutions of undersampled and noisy linear inverse problems. We consider solutions based on compressive sensing and deep learning techniques. Magnetic resonance imaging (MRI) serves as a main motivational example. Starting with compressive sensing techniques and sparse ground truths, a popular solver for such undersampled linear systems is the LASSO. As the distribution of the LASSO is in practice untractable, effective UQ techniques cannot be performed. An adapted estimator, called debiased LASSO, that builds upon the LASSO by adding a correction term, is shown to be asymptotically Gaussian distributed, allowing for the construction of confidence intervals. The requirements, however, do not align with MRI scenarios. Therefore, we extend the debiased LASSO to a class of structured matrices, including subsampled Fourier matrices that describe the MRI setting. We prove asymptotic normality and confirm our theoretical findings with real-world sparse MR images. As sparsity plays a crucial role in this pipeline, we extend the ground truth structure from sparsity with respect to the canonical basis to sparsity in the Haar wavelet and finite gradient domain. Tailoring the debiased LASSO to the former requires the development of a refined sampling of the measurement matrix. In the latter, we tailor the debiasing concept to total variation minimization and evaluate our method with in-vivo MRI data. To bridge theoretical grounded compressive sensing algorithms with data-driven deep learning ones, we focus on unrolled neural networks. While they often perform at a superior level, their architecture is inspired by compressive sensing techniques, allowing for a more sophisticated analysis than purely data-driven methods. We exploit this knowledge and apply the debiasing concept for learned ISTA. To align the confidence intervals based on the asymptotic normality of a debiased estimator with real-world applications, i.e., finite dimensions, we develop a non-asymptotic theory. The main idea is to combine model-based knowledge from the linear inverse problem with a data-driven approach, where we estimate the size of the confidence interval to a certain degree. Thus, our theory applies to a wide range of deep learning estimators such as CNN-based ones or diffusion models. The derived theory is extensively evaluated with real-world MRI data.
Frederik Hoppe (Wed,) studied this question.