Geometric functional analysis, random matrices and applications

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This project seeks to advance methods of geometric functional analysis and to apply them for problems of non-asymptotic random matrix theory and high-dimensional data. One of the main goals of this project is to expand our understanding of non-asymptotic properties of random matrices, in particular, their quantitative invertibility. A related goal is to develop a direct, nonspectral approach to delocalization of eigenvectors of random matrices. Methods of geometric functional analysis may succeed even where spectral methods fail due to an unknown (or nonexisting) limiting spectral distribution. Next, functional analytic and probabilistic methodology will be applied to high-dimensional data. This may result in new geometric approaches to compressed sensing, community detection in networks, and robust principal component analysis. Geometric functional analysis studies fundamental properties of high-dimensional structures. Such structures are ubiquitous in modern applications. The unprecedented volume of data described by large number of parameters prevents many traditional statistical approaches from working. This project will look for new ways to understand, represent, and analyze high-dimensional structures using methods of geometric functional analysis. Specific structures for which the new methodology can be applied include large matrices (e.g., consumer data), networks (e.g., food chains, social networks), signals (e.g., images, audio, video). Conversely, the fundamental challenges of high-dimensional data are likely to open up some new directions of basic research at the intersection of functional analysis, high-dimensional geometry, and probability.