Applied Harmonic Analysis to Non-Convex Optimizations and Nonlinear Matrix Analysis

University Of Maryland, College Park, College Park MD

Investigators

Abstract

This project advances scientific understanding in applied harmonic analysis while promoting teaching, training and learning. The investigator studies two sets of problems, each exploiting redundancy of representation in mathematics and engineering. An example of the first is the problem of estimating simultaneously, from an array of many sensors, the locations of stationary sources and the content of their separate signals, when the sources are not correlated and not all sensors receive information from a given source. Mathematically, this comes down to extracting information from a positive semi-definite covariance matrix that relates signals from sources to receptions by sensors. The existence of unknown blind spots makes this problem challenging. The first thrust concerns the class of positive semi-definite finite trace operators. The aim is to look for decompositions of such operators into sums of rank-one operators that minimize a given criterion. What makes this problem hard is the condition that the rank-ones are also positive semi-definite. It turns out this problem is connected to an open question of Feichtinger in analysis of compact operators with kernels in a modulation space. Additionally, the problem has strong connections to the theory of sparse matrix decomposition, and to array signal processing. The second thrust is related to analysis and optimizations on classes of low-rank positive semi-definite matrices. In particular, this thrust continues the investigator's previous work on the phase retrieval problem and the quantum state tomography problem. Tools from Lipschitz analysis and non-convex optimizations are used throughout this program. Graduate students participate in the research. In addition, the investigator is training them for a globally competitive STEM workforce through his contacts with industry and government labs. The project strengthens existing partnerships with industry while offering opportunities to explore mathematics of real-world applications and to create novel solutions to existing problems. Undergraduate students are encouraged to enter this area of research by participating in existing opportunities under the umbrellas of Research Experience for Undergraduates or Research Interaction Teams. The first problem proposed here relates to the question H. Feichtinger asked in 2004: does the eigen-decomposition of a positive semi-definite trace-class integral operator with kernel in the first modulation space converge in a stronger modulation space-sense? It turns out this question has a negative answer; however, it naturally raises the question of whether a different decomposition of such operators (not necessarily the eigen-decomposition) converges in such a stronger sense. A similar decomposition problem appears in the context of blind source separation. Consider a system composed of many sensors (e.g., antennas, or microphones) and decorrelated wide-sense stationary transmitting sources. The mixing environment has blind spots so that not all sensors receive information from a given source. The problem is to estimate simultaneously location of sources and the source signals, based only on the positive semi-definite covariance matrix. The existence of unknown blind spots is what makes this problem challenging. The second problem of this project refers to quantum state tomography and phase retrieval. Specifically, it asks to estimate low-rank positive semi-definite unit trace matrices from inner products with a fixed set of Hermitian matrices. The project focuses on the class of homotopy methods for matrix recovery. Graduate students participate in the research. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.