Poisson matrix completion and change-point detection
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Statistical signal processing and machine learning are very important in modern science and engineering. Many theories, methods and techniques are developed to help people extract and analyze the hidden information from the Big Data. Big Data has two aspects: 1) the huge number of observations;2) high-dimensionality of data. This dissertation focus on two specific topics that touch the above two aspects. The first topic is low-rank matrix completion. Because many survey data is capable of being represented approximately as a low-rank matrix, this model can apply to many applications such as sensor network, network traffic analysis, sensor localization, recommender systems and natural language processing. Even if much success has been achieved to learn the model with continuous noise, only a few works consider the quantized noise such as Poisson noise, which is applied to many real applications with count data. This motivates us to develop new algorithms and theories for the Poisson Matrix Completion. The second topic is sequential change-point detection. The model is that we observe a sequence of independent signals and would like to detect if an unexpected change happens in the system. This belongs to sequential decision-making problems and we need to decide sequentially whether an alarm of change detected should be raised as we obtain new observations. Due to the nature of sequential decision making, sequential change-point detection has been applied to a large number of engineering applications, such as statistical quality control, financial time series change detection, reliability, surveillance system and system prognostic. In the dissertation, several subtopics are considered, including multi-sensor gradual change detection and robust change-point detection via optimization techniques. For each subtopic, new theories and algorithms are developed.