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 ... read moreValue 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
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