Deep neural networks as the current work horse of artificial intelligence have already been tremendously successful in real-world applications, ranging from science to public life. The area of (medical) imaging sciences has been particularly impacted by deep learning-based approaches, which sometimes by far outperform classical approaches for particular problem classes. However, one current major drawback is the lack of reliability of such methodologies.
In this lecture we will first provide an introduction into this vibrant research area. We will then present some recent advances, in particular, concerning optimal combinations of traditional model-based methods with deep learning-based approaches in the sense of true hybrid algorithms. Due to the importance of explainability for reliability, we will also touch upon this area by highlighting an approach which is itself reliable due to its mathematical foundation. Finally, we will discuss fundamental limitations of deep neural networks and related approaches in terms of computability, and how these can be circumvented in the future, which brings us in the world of quantum computing.
Gitta Kutyniok completed her Diploma in Mathematics and Computer Science in 1996 at the Universitat Paderborn in Germany. She was then employed as a Scientific Assistant and in 2000 received her Ph.D. degree in the area of time-frequency analysis from the same university. In 2001, she spent one term as a Visiting Assistant Professor at the Georgia Institute of Technology. After having returned to Germany, she accepted a position as a Scientific Assistant at the Justus-Liebig-Universitat Giessen. In 2004, she was awarded a Research Fellowship by the DFG-German Research Foundation, with which she spend one year at Washington University in St. Louis and at the Georgia Institute of Technology. She then returned to Germany, completed her Habilitation in Mathematics in 2006 and received her venia legendi. In 2007 and 2008, being awarded one of the highly competitive "Heisenberg Fellowships” by the DFG-German Research Foundation, she spent half a year at each, Princeton University, Stanford University, and Yale University. After returning to Germany in October 2008, she became a full professor for Applied Analysis at the Universitat Osnabrück Gitta Kutyniok was awarded various prizes for both her teaching and research, among which were the "Weierstrass Prize for outstanding teaching of the Universitat Paderborn” in 1998, the "Research Prize of the Universitat Paderborn” in 2003 as well as the "Prize of the University Gießen” in 2006. Just recently, in 2007, she received the prestigious "von Kaven Prize” awarded annually by the DFG-German Research Foundation. Since 2007, she is an Associate Editor for the Journal of Wavelet Theory and Applications, and since 2009, she is a Corresponding Editor for Acta Applicandae Mathematicae. She was a panelist for the NSF in 2008 and serves as a reviewer for the NSF, GIF, NWO, WWTF as well as for over 30 journals. Her research interests include the areas of applied harmonic analysis, numerical analysis, and approximation theory, in particular, sparse approximations, compressed sensing, geometric multiscale analysis, sampling theory, time-frequency analysis, and frame theory with applications in signal and image processing.