Gitta Kutyniok – “The Approximation Power of Deep Neural Networks: Theory and Applications”
“The Approximation Power of Deep Neural Networks: Theory and Applications”
Abstract: Despite the outstanding success of deep neural networks in real-world applications, most of the related research is empirically driven and a mathematical foundation is almost completely missing. The main goal of a neural network is to approximate a function, which for instance encodes a classification task. Thus, one theoretical approach to derive a fundamental understanding of deep neural networks focusses on their approximation abilities.
In this talk we will provide an introduction into this research area. After a general overview of mathematics of deep neural networks, we will discuss theoretical results which prove that not only do (memory-optimal) neural networks have as much approximation power as classical systems such as wavelets or shearlets, but they are also able to beat the curse of dimensionality. On the numerical side, we will then show that superior performance can typically be achieved by combining deep neural networks with classical approaches from approximation theory.
Bio: Gitta Kutyniok currently holds an Einstein Chair in the Institute of Mathematics at the Technische Universität Berlin, a courtesy appointment in the Department of Computer Science and Engineering, an Adjunct Professorship in Machine Learning at the University of Tromsø, and is also the head of the Applied Functional Analysis Group. She received her Diploma in Mathematics and Computer Science as well as her Ph.D. degree from the Universität Paderborn in Germany, and her Habilitation in Mathematics in 2006 at the Justus-Liebig Universität Giessen. From 2001 to 2008 she held visiting positions at several US institutions, including Princeton University, Stanford University, Yale University, Georgia Institute of Technology, and Washington University in St. Louis. In 2008, she became a full professor of mathematics at the Universität Osnabrück, and moved to Berlin three years later. She received various awards for her research such as an award from the Universität Paderborn in 2003, the Research Prize of Giessen and a Heisenberg-Fellowship in 2006, the von Kaven Prize by the DFG in 2007, and an Einstein Chair in 2008. She gave the Noether Lecture at the ÖMG-DMV Congress in 2013 and the Hans Schneider ILAS Lecture at IWOTA in 2016. She also became a member of the BerlinBrandenburg Academy of Sciences and Humanities in 2017, a SIAM Fellow in 2019, and an IEEE Senior Member in the same year. She was Chair of the SIAM Activity Group on Imaging Sciences from 2018-2019 and is Co-Chair of the first SIAM conference on Mathematics of Data Science taking place this year. She is also, for instance, Scientific Director of the graduate school BIMoS at TU Berlin and Chair of the GAMM Activity Groups on Mathematical Signal- and Image Processing and Computational and Mathematical Methods in Data Science, as well as of the MATH+ Activity Group on Mathematics of Data Science. Her main research interests are in the areas of applied harmonic analysis, compressed sensing, highdimensional data analysis, imaging science, inverse problems, machine learning, numerical mathematics, partial differential equations, and applications to life sciences and telecommunication.