چكيده انگليسي :
IAbstract:
The exponential augment and availability of the world’s data have led to the popularity of tensor decomposition and
approximation research. Indeed, encoding these data in a tensor-based format enables us to exploit their innate multidimensional
features. Tensor decomposition has acquired more significance in theoretical and applied mathematics
in the past several years. Tensors play a conspicuous role in diverse applications, such as signal processing, machine
learning, biomedical engineering, neuroscience, computer vision, communication, psychometrics, and chemistry.
They can provide a concise mathematical framework for formulating and solving problems in these fields. However,
there are still impediments to making tensor computations as practical as matrix computations. Recently, higher-order
t-singular value decomposition (Hot-SVD) has come up and given due consideration. The Hot-SVD, also known as
Higher-Order Singular Value Decomposition (HOSVD), is a decomposition that represents a tensor-tensor product
variant. Broadly speaking, Hot-SVD finds innumerable uses in physics and engineering mathematics in tensor-type
data processing. In particular, image and video processing are among the disciplines where abbreviated hot-SVD
algorithms are most critical. In addition, t-product is a type of multiplication between tensors whose framework has
created a leap in theoretical and applied research activities in recent years and has had notable ramifications in matrix
algebra.
To study Hot-SVD, we concentrate on tensor singular value decomposition (t-SVD) and tensor-tensor product in this
thesis. Concerning this matter, we investigate the extension of the t-SVD of third-order tensors to arbitrary order N
tensors, which correspond to order (N-1) tubal tensors. This extension varies from the t-SVD for tensors with orders
more than three. This thesis boils down to follows.
Firstly, this thesis includes some background information on the tensor-tensor product of third-order tensors. Next,
we outline a few fundamental ideas we will require throughout the thesis. These definitions include the Fourier series,
Hilbert matrix, and singular value decomposition. Additionally, we describe the most crucial products, such
as the Hadamard, face-wise, and tensor products, which are necessary to demonstrate the theorems. We also indicate
the Eckart-Young theorem for tubal matrices. Then, well-known concepts and results from matrices, such
as the Kronecker product, are extended to tubal matrices, and the Hot-SVD model for tubular tensors is analyzed.
Moreover, the existence of the Hot-SVD is ascertained, and several properties, including orthogonality and order
related to the HOSVD, are established. Afterward, the truncated Hot-SVD, the sequentially truncated Hot-SVD,
and their error bounds are determined. Likewise, we examine the computational complexity of Hot-SVD, truncated
Hot-SVD, and sequentially truncated Hot-SVD. We consider a numerical example to assess Seq-tr-Hot-SVD, tr-Hot-
SVD, Seq-tr-HOSVD, and tr-HSVD algorithms. This example’s outcome reveals that, in terms of reconstruction
error, tr-Hot-SVD and Seq-tr-Hot-SVD perform better. Since the tubal versions need to execute additional Fourier
transforms, seq-tr-HOSVD is the fastest based on CPU time; nevertheless, tr-HOSVD and seq-tr-HOSVD are slower
than seq-tr-HOSVD and seq-tr-HOSVD. Furthermore, compared to tr-Hot-SVD, seq-tr-Hot-SVD is roughly two to
four times faster. Finally, the appendix concludes with an overview of image processing, which is consequential in
many situations involving artificial intelligence, machine learning, and pattern recognition.
Isfahan University
