In the past decade, there has been a surge of interest in processing and analyzing mixture matrix-valued models arising in many scientific fields such as genomics, neuroimaging, and social network analysis. Owing to the ultra-large dimensions of the observed matrices, recent works have assumed certain types of low-rankness for the underlying parameters of interest. Such low-rankness assumption is also well-motivated in practice, as many datasets exhibit low-rank structure due to shared latent factors or sparse interactions among variables. Unlike the promising empirical performance of various mixture matrix-valued models, their theoretical properties were less justified. This motivates our first part of the thesis, focusing on fundamental limits of estimation in low-rank Gaussian mixtu...[ Read more ]

In the past decade, there has been a surge of interest in processing and analyzing mixture matrix-valued models arising in many scientific fields such as genomics, neuroimaging, and social network analysis. Owing to the ultra-large dimensions of the observed matrices, recent works have assumed certain types of low-rankness for the underlying parameters of interest. Such low-rankness assumption is also well-motivated in practice, as many datasets exhibit low-rank structure due to shared latent factors or sparse interactions among variables. Unlike the promising empirical performance of various mixture matrix-valued models, their theoretical properties were less justified. This motivates our first part of the thesis, focusing on fundamental limits of estimation in low-rank Gaussian mixture model (LrMM) and revealing a statistical-to-computational gap. Another related yet more practical task in LrMM is clustering. In the second part of the thesis, we develop an modified Lloyd’s algorithm that utilizes the planted low-rankness with optimal error rate. The algorithm has been extensively evaluated on both synthetic and real-world datasets with appealing performance. The third part of the thesis shifts focus to network data. We propose a flexible mixture model concerning multi-layer networks, together with a tensor-based algorithm to consistently recovering local/global communities of nodes as well as labels of networks, and its application on real datasets yields new and interesting findings.