All ETDs from UAB

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.

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