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How to handle large multi-dimensional datasets such as hyperspectral images and video information both efficiently and effectively plays an important role in big-data processing. The characteristics of tensor low-rank decomposition in recent years demonstrate the importance of capturing the tensor structure adequately which usually yields efficacious approaches. In this dissertation, we first aim to explore the tensor singular value decomposition (t-SVD) with the nonconvex regularization on the multi-view subspace clustering (MSC) problem, then develop two new tensor decomposition models with the Bayesian inference framework on the tensor completion and tensor robust principal component analysis (TRPCA) and tensor completion (TC) problems. Specifically, the following developments for multi-dimensional datasets under the mathematical tensor framework will be addressed. (1) By utilizing the t-SVD proposed by Kilmer et al. \cite{kilmer2013third}, we unify the Hyper-Laplacian (HL) and exclusive $\ell_{2,1}$ (L21) regularization with Tensor Log-Determinant Rank Minimization (TLD) to identify data clusters from the multiple views' inherent information. Whereby the HL regularization maintains the local geometrical structure that makes the estimation prune to nonlinearities, and the mixed $\ell_{2,1}$ and $\ell_{1,2}$ regularization provides the joint sparsity within-cluster as well as the exclusive sparsity between-cluster. Furthermore, a log-determinant function is used as a tighter tensor rank approximation to discriminate the dimension of features. (2) By considering a tube as an atom of a third-order tensor and constructing a data-driven learning dictionary from the observed noisy data along the tubes of a tensor, we develop a Bayesian dictionary learning model with tensor tubal transformed factorization to identify the underlying low-tubal-rank structure of the tensor substantially with the data-adaptive dictionary for the TRPCA problem. With the defined page-wise operators, an efficient variational Bayesian dictionary learning algorithm is established for TPRCA that enables to update of the posterior distributions along the third dimension simultaneously. (3) With the defined matrix outer product into the tensor decomposition process, we present a new decomposition model for a third-order tensor. The fundamental idea is to decompose tensors mathematically in a compact manner as much as possible. By incorporating the framework of Bayesian probabilistic inference, the new tensor decomposition model on the subtle matrix outer product (BPMOP) is developed for the TC and TRPCA problems. Extensive experiments on synthetic data and real-world datasets are conducted for the multi-view clustering, TC, and TRPCA problems to demonstrate the desirable effectiveness of the proposed approaches, by detailed comparison with currently available results in the literature.

Timely Fault Detection and Diagnosis in complex manufacturing systems is critical to ensure safe and effective operation of plant equipment. Process fault is defined as a deviation from normal process behavior, defined within the limits of safe production. The quantifiable objectives of Fault Detection include achieving low detection delay time, low false positive rate, and high detection rate. Once a fault has been detected pinpointing the type of fault is needed for purposes of fault mitigation and returning to normal process operation. This is known as Fault Diagnosis. Data-driven Fault Detection and Diagnosis methods emerged as an attractive alternative to traditional mathematical model-based methods, especially for complex systems due to difficulty in describing the underlying process. A distinct feature of data-driven methods is that no a priori information about the process is necessary. Instead, it is assumed that historical data, containing process features measured in regular time intervals (e.g., power plant sensor measurements), are available for development of fault detection/diagnosis model through generalization of data. The goal of my research was to address the shortcomings of the existing data-driven methods and contribute to solving open problems, such as: 1) decentralized fault detection and diagnosis; 2) fault detection in the cold start setting; 3) optimizing the detection delay and dealing with noisy data annotations. 4) developing models that can adapt to concept changes in power plant dynamics. For small-scale sensor networks, it is reasonable to assume that all measurements are available at a central location (sink) where fault predictions are made. This is known as a centralized fault detection approach. For large-scale networks, decentralized approach is often used, where network is decomposed into potentially overlapping blocks and each block provides local decisions that are fused at the sink. The appealing properties of the decentralized approach include fault tolerance, scalability, and reusability. When one or more blocks go offline due to maintenance of their sensors, the predictions can still be made using the remaining blocks. In addition, when the physical facility is reconfigured, either by changing its components or sensors, it can be easier to modify part of the decentralized system impacted by the changes than to overhaul the whole centralized system. The scalability comes from reduced costs of system setup, update, communication, and decision making. Main challenges in decentralized monitoring include process decomposition and decision fusion. We proposed a decentralized model where the sensors are partitioned into small, potentially overlapping, blocks based on the Sparse Principal Component Analysis (PCA) algorithm, which preserves strong correlations among sensors, followed by training local models at each block, and fusion of decisions based on the proposed Maximum Entropy algorithm. Moreover, we introduced a novel framework for adding constraints to the Sparse PCA problem. The constraints limit the set of possible solutions by imposing additional goals to be reached trough optimization along with the existing Sparse PCA goals. The experimental results on benchmark fault detection data show that Sparse PCA can utilize prior knowledge, which is not directly available in data, in order to produce desirable network partitions, with a pre-defined limit on communication cost and/or robustness. / Computer and Information Science