18.06.2018 14:00 Judith Wewerka (TUM):
Near-optimal data-driven l^1-regularization (Master Thesis presentation)MI 03.10.011 (Boltzmannstr. 3, 85748 Garching)

Solving underdetermined linear systems of equations by l^1-regularization is a common approach. The amount of regularization is controlled by a regularization parameter. Even though there exist various techniques for choosing this parameter, it is still a relevant challenge. In this thesis, a data-driven approach is proposed, which is neither relying on knowledge of the solution nor on the noise level. The idea is to estimate the regularization parameter of LASSO by a greedy solution computed by OMP. We give explicit error bounds for the vectors reconstructed by the different algorithms and show theoretically that by an optimal choice of the regularization parameter LASSO and OMP can achieve the same error. The numerical results are even more promising: we find scenarios where LASSO, with an estimated parameter, outperforms OMP. Furthermore, we implement an image reconstruction from noisy and undersampled data in MATLAB. Here, LASSO using a regularization parameter chosen by our proposed approach, reconstructs with a smaller error than OMP.