A Novel Algebraic Framework for Processing Multidimensional Data: Theory and Application
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
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