High-dimensional data today plays an increasingly important role in most computer-based measurements and analyses of real-life problems. This development is due to several technological advances, such as in increased physical measurements and storage capabilities, and multiphysics computer simulations of complex phenomena. Analyses of such data can reveal complex interactions between entities in a process in Nature, or in man-made environments such as Instagram or a Google search.
At the same time we are struggling with extracting useful information and derive predictive models. Data-driven modelling is an emerging and challenging area in applied mathematics with an enormous potential, especially if successfully combined with other branches of science, like computer science, engineering, or biomedical computing.
Motivated by the increased demand of robust and predictive methods, we will in this project develop powerful mathematical and numerical machinery to explore new applications in fields such as tractable and robust automatic learning of functions and data structures in high dimension from the minimal number of observed samples. The approach we propose is to obtain lower-dimensional representations of the data and their geometry, to perform the approximation with significantly reduced complexity under realistic assumptions.
The proposed project will be based on both a rigorous theoretical analysis as well as on implementation and numerical tests of the corresponding algorithms to demonstrate their robustness and performance. The realisation of the project will require the deployment of various mathematical techniques, consisting of theory of inverse and ill-posed problems, learning and approximation theories, differential geometry, harmonic analysis, and sparse optimisation. Finally, we will also address problems in computational biomedicine, such as encountered in cardiac modeling, and gene expressions in bioinformatics.
The three-year project, funded by the Research Council of Norway as part of the FRINATEK and IKTPLUSS programs, will be conducted under the supervision of Dr. Valeriya Naumova, in closed collaboration with the research groups at Technical University Munich (Prof. Dr. M. Fornasier), Duke University (Prof. Dr. M. Maggioni), University College London (Prof. Dr. M. Pontil), University of Innsbruck (Dr. K. Schnass), and Norwegian University of Science and Technology (Assoc. Prof. Dr. M. Grasmair).
Research Council of Norway (FRINATEK, IKTPLUSS)
Dr. Valeriya Naumova