FRG: Collaborative Research: Mathematical and Statistical Analysis of Compressible Data on Compressive Networks

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Large-scale high-dimensional data sets are becoming ubiquitous in modern society, particularly in the areas of physical, biomedical, and social applications. This focused research group (FRG) will address the foundational challenges, both computational and theoretical, arising in the analysis of high-dimensional data by leveraging its compressible features. Discovering such compressible features is a major challenge in data analysis, which the team of investigators will approach using hierarchical decompositions derived from spectral, statistical, and algebraic geometric analysis of data. In contrast to interpolation-based methods, such as deep neural networks which are often difficult to interpret, the group will construct optimally defined compressive networks, specifically tailored to such compressible features. Doing so will enable an accurate and efficient extraction and manipulation of sparse representations of high-dimensional data in an inherently interpretable manner. For instance, one focus of the project is to extend the binary expansion testing methods developed by members of the group, which have shown promise in both statistical power and computational complexity in low-dimensional settings. A high-dimensional generalization of binary expansion testing would, in turn, enable the direct application to selecting personalized medical treatment plans based on increasingly complex data sets. The FRG investigators will collaborate across the disciplines of mathematical analysis, data science, statistics, and computation, as well as across institutions. The specific goals of this project include generalizing classical concepts of "compressible" features using ideas from spectral theory, algebraic geometry, energy and optimization, and network interactions. This will lead to a deeper understanding of the mathematical and statistical foundations of compressible high-dimensional data sets on compressive networks. Using newly developed compressible features, the FRG team will then design and develop accurate and efficient computational tools for large-scale high-dimensional data sets. All the work to be done will be aimed at collaborating directly with application domain scientists to enhance the efficacy of the proposed methods. The FRG investigators will also jointly mentor graduate and undergraduate students, who will then have the benefits of training across disciplines and access to a variety of ideas and tools in complementary and integrative research areas.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.