%0 PDF
%T DESIGN AND APPLICATION OF TENSOR DECOMPOSITIONS TO PROBLEMS IN MODEL AND IMAGE COMPRESSION AND ANALYSIS
%A Zhang, Jiani.
%D 2017-06-29T09:04:37.223-04:00
%8 2017-07-07
%R http://localhost/files/2b88qq41w
%X 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 computation, 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.
%[ 2022-10-11
%9 Text
%~ Tufts Digital Library
%W Institution