DESIGN AND APPLICATION OF TENSOR DECOMPOSITIONS TO PROBLEMS IN MODEL AND IMAGE COMPRESSION AND ANALYSIS
Zhang, Jiani.
2017
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Abstract: Tensor algebra and tensor computations have gained more and more attention in recent years due to their ability to handle and explore large-scale, high-dimensional datasets. In this thesis, we present four novel tensor-based methods in the fields of randomized algorithms, dynamical systems, image processing, and video processing. In the first chapter, we introduce the history of tensor c... read moreomputation, discuss well-known tensor operators and decompositions, and demonstrate our motivations to focus on the t-product-based operators and decompositions designed by Professors Kilmer and Martin \cite{2011kilmer}. Then, in Chapter \ref{chp:rt-svd}, we design a method called randomized tensor singular value decomposition that can produce a factorization with similar properties to the tensor SVD (t-SVD) but that is more computationally efficient on very large datasets. We present the details of the algorithm and the theoretical results, and we provide numerical results on two public facial recognition datasets. Chapter \ref{chp:pod} addresses the problem of model reduction on dynamical systems. We investigate the proper orthogonal decomposition (POD) method, compare the approximation errors obtained from truncated SVD and truncated tensor SVD in theory, and provide an effective projector for the POD method using truncated tensor SVD. Chapter \ref{chp:tensor-MBD} and Chapter \ref{chp:video} are both devoted to optimization-related problems. In Chapter \ref{chp:tensor-MBD}, for the multi-frame blind deconvolution optimization model, we design a method to select the most representative frames that is less heuristic in nature than current methods. In Chapter \ref{chp:video}, we use tensor operators to model the video resolution enhancement problem and leverage the tensor nuclear norm as a regularization term to minimize the rank of its solution.
Thesis (Ph.D.)--Tufts University, 2017.
Submitted to the Dept. of Mathematics.
Advisor: Misha Kilmer.
Committee: James Adler, Lior Horesh, and Xiaozhe Hu.
Keyword: Mathematics.read less - ID:
- 2b88qq41w
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