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ACMD Seminar: Global Lipschitz Analysis in Inverse Problems

Professor Radu Balan
Applied Mathematics, University of Maryland and Center for Scientific Computation and Mathematical Modeling

Wednesday, May 30, 2018 15:00 - 16:00
Building 227 Room A202

Wednesday, May 30, 2018 13:00 - 14:00
Building 1 Room 4058

Host: Isabel Beichl

Abstract: This talk is devoted to Lipschitz analysis on several inverse problems in applied harmonic analysis. Specifically we analyze the phase retrieval problem, the low-rank quantum tomography, the compressive sampling, and sparse blind signal separation. In each case, for appropriate metric space structures, the forward nonlinear map is bi-Lipschitz. The problem is whether a globally Lipschitz left inverse reconstruction map exists, and if so, to find its smallest Lipschitz constant. Two extension principles are analyzed: the Whitney-McShaun (also known as the nonlinear Hahn-Banach) extension, and the Kirszbraun extension. In the phase retrieval problem, the Lipschitz extension is possible with low cost in the optimal constant; in the low-rank quantum tomography problem, Lipschitz extension is not possible due to topological obstructions; in the compressive sampling, Lipschitz extension is possible with dimension dependent bounds; in the sparse blind signal separation, the answer is not known.

Bio: Radu Balan received the Bachelor of Engineering degree in Electrical Engineering from the Polytechnic Institute of Bucharest,  Romania, in 1992, the Bachelor of Science in Physics from the University of Bucharest, Romania, in 1994, and the Ph.D in Applied Mathematics from Princeton University, Princeton, NJ, in 1998. After an 8-year appointment with the Siemens Corporate Research in Princeton, NJ, he moved in 2007 to the University of Maryland, College Park, where he is a Professor of Applied Mathematics with a joint appointment at the Center for Scientific Computation and Mathematical Modeling. His research interests are in applied harmonic analysis, redundant representations, machine learning and nonlinear analysis. He is currently a member on the editorial board of the Journal of Applied and Computational Harmonic Analysis (since 2005), and an Associate Editor of the IEEE Transactions on Information Theory (2017-2019).

Note: Visitors from outside NIST must contact Cathy Graham; (301) 975-3800; at least 24 hours in advance.

Created May 10, 2018, Updated November 15, 2019