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Thesis : Convex Optimization Algorithms and Recovery Theories for Sparse Models in Machine Learning, Bo Huang

- Sparse modeling is a rapidly developing topic that arises
frequently in areas such as machine learning, data analysis and signal
processing. One important application of sparse modeling is the recovery
of a high-dimensional object from relatively low number of noisy
observations, which is the main focuses of the Compressed Sensing,
Matrix Completion(MC) and Robust Principal Component Analysis (RPCA) .
However, the power of sparse models is hampered by the unprecedented
size of the data that has become more and more available in practice.
Therefore, it has become increasingly important to better harnessing the
convex optimization techniques to take advantage of any underlying
"sparsity" structure in problems of extremely large size. This thesis
focuses on two main aspects of sparse modeling. From the modeling
perspective, it extends convex programming formulations for matrix
completion and robust principal component analysis problems to the case
of tensors, and derives theoretical guarantees for exact tensor recovery
under a framework of strongly convex programming. On the optimization
side, an efficient first-order algorithm with the optimal convergence
rate has been proposed and studied for a wide range of problems of
linearly constraint sparse modeling problems.

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