Statistical inference of high-dimensional data is crucial for science and engineering. Such high-dimensional data are often structured. For example, they can be data from a certain manifold or a large network. Motivated by the problems that arise in recommendation systems, power systems, and social media, etc., this dissertation aims to provide statistical modeling for such problems and perform statistical inferences. This dissertation focus on two problems. (i) statistical modeling for smooth manifold and inferences for the corresponding characteristic rank; (ii) detection of change-points for sequential data in a network. For the first topic. We start with the rank selection problem in the matrix completion problem. We addressed the problem of rank identifiability in minimum rank matrix completion problem and proposed a statistical model for the low-rank matrix approximation problem. We then generalize the problem to a more general smooth manifold. For the second topic. We study the problem of cascading failure motivated by the study of the power system. We proposed a model for failure propagation and a fast algorithm to perform the test procedure of detecting the cascading failure. The other problem we study in change-points detection is to detect the change of event data. We use the multivariate Hawkes process to capture the self and cross excitation between the events and proposed a test procedure base on scan score statistics.Ph.D
