AF: Small: Geometric Data Processing and Analysis via Light-weight Structures

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In the current era of information, large amounts of complex data are routinely generated across science and engineering. Geometric and topological data analysis is a major component involved in organizing, analyzing, as well as utilizing the huge amount of data generated. As data becomes more complex and high dimensional, it can be computationally formidable to compute and maintain even local structures from data. Hence it has become crucial to identify and use light-weight structures that can be efficiently computed, maintained and/or manipulated; while at the same time are also sufficiently informative. This project will focus on two such light-weight structures: the Reeb graph and the Gaussian-weighted graph Laplacian. Both of them have already been extensively used for various data analysis applications. Nevertheless, there are many theoretical and algorithmic questions about them that need to be addressed to better understand their behavior, their potential, as well as to further their practical usage. This project aims to address these issues, develop a solid theoretical foundation as well as efficient algorithms for their practical usage, and investigate novel geometric processing applications based on these two objects. Data analysis is a fundamental problem in computational science, ubiquitous in a broad range of application fields in science and engineering. The intellectual value of this project lies not only in developing novel computational geometry algorithms and techniques, but also in transferring the results and insights to practical applications. Results from this project, especially algorithms and software, will be disseminated to a large scientific community so as to potentially produce a broader impact in scientific and engineering endeavors.