Advisory Committee Chair
Carmeliza Navasca
Advisory Committee Members
Brendan Ames
Ian Knowles
Da Yan
Guo-Hui Zhang
Document Type
Dissertation
Date of Award
2019
Degree Name by School
Doctor of Philosophy (PhD) College of Arts and Sciences
Abstract
Tensor decompositions are higher-order analogues of matrix decompositions which have applications in data analysis, signal processing, machine learning and data mining. One of the most challenging problems in the tensor decomposition area is to approximate the rank of a given tensor. Unlike the matrix case there is no simple formula to bound the rank of a tensor. In fact, finding the exact rank of a tensor is an NP hard problem. In this thesis we formulate the tensor rank estimation of a tensor as an optimization problem and estimate the rank via $\ell_1$ minimization. We propose a numerical iterative method based on the proximal alternating minimization algorithm and discuss the required conditions for the global convergence of the algorithm. The performance of our algorithm is tested on several types of data such as randomly generated tensors, RGB images and surveillance videos in order to separate the background and foreground of them. In addition, this thesis studies the block sampling of the alternating least squares technique (ALS) and proposes an effective method for the CP decomposition of tensors. The method is tested on randomly generated data as well as real data. The proposed method converges faster that ALS in some cases when there is a presence of swamp. In addition, it requires less computations in each iteration. Also, the application of CP decomposition in image compression is provided.
Recommended Citation
Goudarzi Karim, Ramin, "Tensor Decompositions and Rank Approximation of Tensors with Applications" (2019). All ETDs from UAB. 1771.
https://digitalcommons.library.uab.edu/etd-collection/1771