Computing reliable solutions to inverse problems is important in many applications such as biomedical imaging, computer graphics, and security. Regularization by incorporating prior knowledge is needed to stabilize the inversion process. In this talk, we develop a new framework for solving inverse problems that incorporates probabilistic information in the form of training data. We provide theoretical results for the underlying Bayes risk minimization problem and discuss efficient approaches for solving the associated empirical Bayes risk minimization problem. Various constraints can be imposed to deal with large-scale problems. Here we describe methods for computing optimal spectral filters, for cases where the SVD is available, and methods for computing an optimal low-rank regularized inverse matrix, for cases where the forward model is not known.
This is joint work with Matthias Chung (Virginia Tech) and Dianne O’Leary (University of Maryland, College Park).