Techniques in network embedding and Gaussian comparison for high-dimensional statistics

This dissertation consists of research on two high-dimensional statistical problems. In the first part of the dissertation, we study Gaussian comparison which is an important technique for comparing distributions and functionals of Gaussian random variables. We derive a Gaussian comparison result based on a smart-path argument. We show the significance of this result by an application to a problem of maximal correlations in high dimensions.

In the second part of the dissertation, we focus on the problem of community detection on networks using node embedding methods. In recent decades, network data sets containing millions and billions of nodes have become available. This has necessitated the development of scalable methods for their analysis. One such class of methods are methods for node embedding. Node embedding methods encode nodes of a network in a low-dimensional Euclidean space which allows one to use well-known methods for Euclidean spaces for network analysis. In this dissertation we study the problem of community detection using two well-known node embedding methods: DeepWalk and node2vec. We describe the network sparsity regimes when the k-means algorithm applied to the node embeddings detects communities for graphs generated from the stochastic block model, and when such an approach might fail. We also describe how increasing the non-backtracking parameter in the node2vec method leads to provable improvements in community detection.

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