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Novel Statistical Methods for High-Dimensional Data: Adversarial Subspace Learning and Quickest Detection of Community
This thesis studies different statistical methods for analyzing high-dimensional data. The first chapter of the thesis focus on the domain of robust subspace learning under the influence of adversarial attacks. By analyzing the adversarial projection risk when the data follow the Spiked Covariance model, we advocate the employment of the empirical risk minimization (ERM) approach in order to ascertain the optimal robust subspace. We derive a non-asymptotic upper bound of adversarial excess risk, which insinuates that the ERM estimator exhibits a close approximation to the optimal robust adversarial subspace. To obtain the ERM under the general spiked covariance model which is computationally intractable, we suggest minimizing an upper bound of the empirical risk. The second chapter of the thesis focus on the problem of detection of community structures which is a pivotal challenge in network analysis. In this research, we employ the Erdős-Rényi model in conjunction with the bisection stochastic block model (SBM) to delineate the pre-change and post-change distributions of the network, respectively. We put forth an advanced monitoring procedure predicated upon the exploitation of the number of k-cycles present in the graph. We derive the asymptotic detection properties of the proposed technique, contingent upon the knowledge of all parameters. To tackle scenarios wherein parameters remain unknown, we devise a generalized likelihood ratio (GLR) type detection procedure and an adaptive CUSUM type detection procedure. In the third chapter, we develop an extension to the general stochastic block model when the community sizes are unequal and unknown for us.
Statistics|Computer science|Applied Mathematics
Sha, Fei, "Novel Statistical Methods for High-Dimensional Data: Adversarial Subspace Learning and Quickest Detection of Community" (2023). ETD collection for University of Nebraska - Lincoln. AAI30488606.