A Novel Algebraic Framework for Processing Multidimensional Data: Theory and Application
Zhang, Zemin.
2017
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Abstract: Tensor related analysis and applications are more and more popular
in computer vision, machine learning, data mining, psychometrics, signal processing and
other areas. In this thesis, we first investigate a recently proposed tensor algebraic
framework, in which one can obtain a factorization for multidimensional data, referred to
as the tensor-SVD (t-SVD) as similar to the Singular V... read morealue Decomposition (SVD) for
matrices. t-SVD results in a notion of rank referred to as the tubal-rank. Using this
approach, we consider the problem of sampling and recovery of 3-D arrays with low
tubal-rank. We show that by solving a convex optimization problem, which minimizes a convex
surrogate to the tubal-rank, one can guarantee exact recovery with high probability as long
as the number of samples is of the order O(rnk log(nk)) given a tensor of size n x n x k
with tubal-rank r. The conditions under which this result holds are similar to the
incoherence conditions for low-rank matrix completion under random sampling. The difference
is that we define incoherence under the algebraic set-up of t-SVD, which is different from
the standard matrix incoherence conditions. We also compare the numerical performance of
the proposed algorithm with some state-of-the-art approaches on real-world datasets. After
that, we discuss the t-SVD based robust PCA methods, in both batch and online manner.
Applications on image denoising and fusing cloud-contaminated satellite images demonstrate
that the proposed method shows superiority in both convergence speed and performance
compared to the state-of-the-art approaches. In the end, a new dictionary learning
algorithm for multidimensional data is proposed. Unlike most conventional dictionary
learning methods which are derived for dealing with vectors or matrices, our algorithm,
named K-TSVD, learns a multidimensional dictionary directly via t-SVD. We propose to extend
K-SVD algorithm used for 1-D data to a K-TSVD algorithm for handling 2-D and 3-D data. Our
algorithm, based on the idea of sparse coding (using group-sparsity over multidimensional
coefficient vectors), alternates between estimating a compact representation and dictionary
learning. We analyze our K-TSVD algorithm and demonstrate its result on video completion
and video/multispectral image denoising.
Thesis (Ph.D.)--Tufts University, 2017.
Submitted to the Dept. of Electrical Engineering.
Advisor: Shuchin Aeron.
Committee: Eric Miller, Dehong Liu, and Misha Kilmer.
Keyword: Electrical engineering.read less - ID:
- 9g54xw08k
- Component ID:
- tufts:20679
- To Cite:
- DCA Citation Guide EndNote
