NSF-SNSF: Functional data analysis for complex systems

University Of North Carolina At Chapel Hill, Chapel Hill NC

Investigators

Abstract

Networks play a major role in many disciplines, both as the primary medium of interest, for instance, the flow of (mis)-information in social networks, or unobserved relationships describing developmental trajectories in cells in individuals, or as a fundamental ingredient in representing high dimensional data in sophisticated multi-step machine learning pipelines. While tremendous advancements have been made in the formulation and application of network-driven techniques on data, the main aim of this project is to provide a theoretical understanding of such models and the accuracy of ensuing scientific conclusions. The project will focus on two major sub-domains, (1) understanding multilayer network data, e.g., network-valued data on a single individual over multiple time points or multi-population data points across different tasks as well as understanding the time evolution of such systems and (2) developing mathematical techniques to understand properties of a major class of techniques used to analyze high dimensional data, namely Gaussian graphical models. This project also provides research training opportunities for graduate students. The project is focused on two major areas of statistical methodology related to functional data analysis for complex systems: (I) Optimal transport for multilayer networks and trajectory inference for complex systems and (II) Continuum scaling limits in Graphical models. In the first domain, the PIs will develop statistically principled techniques for network summarization, clustering, and extraction of principle directions of variation building on Gaussian process optimal transport techniques from functional data analysis, and specifically the representation of Procrustes metrics on covariance operators via the Wasserstein distance between corresponding Gaussian processes. Related to this first domain, motivated by single-cell RNA-seq and network neuroscience, the project will develop mathematical techniques to understand optimal transport-based methods for registration (time synchronization) and supervised learning tasks, including network clustering, after quotienting out the underlying developmental trajectory. Next, driven by areas such as gene-expression data from cancer genomics, the main goal for the second theme is the study of high-dimensional data with underlying dependency structures modulated by a latent network connecting the features. Mathematical techniques that will be developed include (a) Thresholding pipelines from covariance and correlation matrices and local weak convergence of associated objects to limit infinite structures and corresponding implications for thresholding schemes; (b) Hierarchical Representation learning for complex systems and their connection to convergence to continuum scaling limits via connections between linkage clustering and thresholding; (c) Structured alternatives, penalized estimation and limiting distributions of random adjacency matrices including localization phenomena for eigenvectors and their use in hub-detection. This collaborative U.S.-Swiss project is supported by the U.S. National Science Foundation (NSF) and the Swiss National Science Foundation (SNSF), where NSF funds the U.S. investigator and SNSF funds the partners in Switzerland. 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.