Applied Harmonic Analysis Methods for Non-Convex Optimizations and Low-Rank Matrix Analysis

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This project promotes the progress of science by expanding the area of applied harmonic analysis both in mathematics and in applications. The research addresses complex optimization problems that are intractable by current super-computers. The projects broaden the two-way communication between mathematics on one side and engineering and computer science on the other side while promoting teaching, training, and learning. The PI will train graduate students for a globally competitive STEM force through internships in industry and government labs. The project increases the existing partnerships with industry while offering opportunities to explore mathematics of real-world applications and to create novel solutions to existing problems.The project contains two thrusts. The first thrust develops homotopic methods for non-convex optimizations. In particular, the project will focus on low-rank matrix estimation, such as the case in the phase retrieval problem, and quadratic assignment problems, as in graph matching problems. The homotopic method extends the original search space (the phase space) by one continuous parameter that trades between the target non-convex objective function and a carefully chosen convex penalty term. A path tracker is initialized at the global optimizer and gradually evolved towards the global optimum of the non-convex objective function. The research will obtain guarantees of optimality for this method. The second thrust relates to geometric and functional analysis of low-rank positive semi-definite matrices. The program studies various metrics and Lipschitz embeddings of these metric spaces. Preliminary results show a rich and complex collection of metrics on this semi-algebraic variety. In particular, one such measure is related to optimal expansions into sums of nonnegative rank-one matrices. It turns out this decomposition is directly related to the analysis of compact integral operators with kernels in certain modulation spaces. These results will be extended to finite-dimensional settings.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.