Inverse problems naturally arise in many scientific settings, and the study of these problems has been crucial in, for example, the development of medical imaging technology. In an inverse problem, the goal is to reconstruct an image from its measurements, using knowledge of the measurement process. The inversion of the measurement process is usually ill-posed and needs to be stably approximated to ensure high-quality reconstructions if the measurements are noisy. A well-established paradigm for doing so is the variational regularisation approach, in which images are estimated by minimising a functional that trades off the fit to the measurements and the fit to a model of the prior knowledge about the true image.
As impressive as many recent developments in machine learning are in terms of their performance for certain tasks when trained correctly, their theoretical analysis — in particular in terms of their robustness to outliers and their error control and prediction — is mostly missing. One of the cornerstones of the Institute will be the development of rigorous learning methodologies that are accessible by mathematical and statistical analysis techniques, resulting in controllable guarantees on the decision-making of smart machines. Recent developments in this context include so-called bi- (or multi-) level optimisation approaches http://arxiv.org/abs/1505.02120
whose mathematical analysis and computational solution are key for proposing both adaptive and reliable analysis techniques. Applications of interest include adaptive image analysis such as image classification, segmentation and enhancement, all the way to inverse problems in industrial and medical imaging, seismics and inference for high-dimensional data.
Joint work with Elena Celledoni (NTNU) and Brynjulf Owren (NTNU).