Project B2: Inverse problems in exponential families: multiresolution data fidelities
In this project, we develop and investigate multiscale variational estimators (MVE) for statistical inverse problems. These are given by minimization of a (typically) convex target functional within a multiresolution constraint. The latter is formulated in ensembles (dictionaries) of probe functionals which act as local statistical tests. Examples of such target functionals are Besov and BV norms. We envision to extend these multiscale estimators to variational estimators with adversarial data-ﬁdelities. The latter technique from the ﬁeld of machine learning has been used among others for sampling from very high dimensional densities, a challenging and important issue. One goal of this project will be to extend our approach to this problem. Furthermore, we plan to combine our techniques with the statistical analysis of deep neural networks, to obtain risk bounds for these estimators. In a broader sense, the ultimate goal for this PhD project is to study the application of neural networks to statistical inverse problems. Our approach is to combine the ﬂexibility of neural networks with problem speciﬁc multiscale dictionaries, thereby bringing the structure of inverse problems and the statistical optimality of dictionary methods into the general setting of neural networks.
Application: various biological and medical imaging problems