Multiscale random matrices, inference and learning in high dimensions

In several areas of scientific investigation, such as genomics, astrophysics, and network traffic, traditional modeling tools are inadequate for handling the vast amounts of data generated by modern technology. This project will bring together a team of collaborators to tackle the challenge of developing new methodologies for analyzing complex and multiscale data. The group is both multidisciplinary (mathematics, statistics, and signal processing) and international (U.S. and France). The driving area of application is the modeling of brain dynamics and connectivity, as well as its repercussions for our understanding of neurological functions and disorders. By training both graduate and undergraduate students, this project will contribute to preparing the U.S. workforce for jobs that require knowledge of some of the latest trends in data science, including high-dimensional statistics, machine learning, and AI. The project will develop a mathematical framework for the fractal modeling of high-dimensional data through the lens of multiscale random matrices (MRMs). MRMs offer a general approach to large-scale, complex stochastic dynamics. They are particularly suited to the analysis of systems where the number of time series (namely, the dimension) is comparable to the number of observations. In the framework of MRMs, newly uncovered universality properties of random matrix statistics will guide the construction of robust asymptotic results. In addition, MRMs will underpin the development of graph-based methodology for high-dimensional, nonstationary systems. In the context of multi-sample problems, the project will further add to the recent stream of literature on random matrix theory applied to statistical learning by developing data-scientific methodologies that are intrinsically multiscale and robust. 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.