Multivariate Analysis for Samples of Networks

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Data collected in the form of networks, often with weighted edges, have become increasingly common in practice, but classical statistical tools do not apply to such data. This project will develop a statistical toolbox of network-aware methods for several common statistical analyses. The main motivating application is neuroimaging, which allows collecting brain connectivity network data from human subjects. These methods can be used to identify brain connectivity patterns associated with disorders, to estimate normal brain development trajectories and deviations from it for children and adolescents, to study changes associated with normal and abnormal aging in older subjects, and to find new subtypes of known disorders. The methods will be developed and extensively tested in close collaboration with neuroscientists.This project develops network analogues for common multivariate analysis tools for vector-valued data, such as estimating the mean, classification, clustering, and principal component analysis. The overarching theme is developing network-aware methods by striking the balance between collapsing the networks to a few global summary measures and treating them as a long single vector of edge weights, with the goal to obtain methods that are not only accurate, but also scientifically interpretable. The technical challenges in developing such methods arise, broadly speaking, from the need to impose high-level network structure, such as communities, onto low-level network features, such as individual edge weights. Addressing these challenges will involve sophisticated tools from modern random matrix theory, structured penalties, advanced optimization techniques, and computationally efficient algorithms, to be developed for each of the new network analysis tools proposed. The close collaboration with neuroscientists and psychiatrists will ensure that the statistical tools developed are thoroughly tested and vetted in the neuroimaging community. Project results will be widely disseminated through publications, presentations, and software packages, in both statistical and neuroimaging venues, and are expected to raise the current standards for statistical network analysis in the field of brain connectomics. The project will also train graduate students in an important emerging area of modern statistics and help them develop advanced computing and interdisciplinary collaboration skills.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.