Low-rank Tensor Completion: Algorithm Development and Its Application in Signal Processing


Student thesis: Doctoral Thesis

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  • Longting HUANG

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Award date25 Apr 2016


Low-rank tensor completion has many important applications, such as in recommendation systems, image inpainting, video decoding, computer vision, machine learning, data mining, array signal processing, seismic data reconstruction, and intelligent transportation systems. As an extension of low-rank matrix completion to higher-dimensional space, low-rank tensor completion utilizes the low-rank structure embedded into the observed tensor to recover its missing values with the knowledge of a subset of its entries. Many factors contribute to the aforementioned missed data, such as when receiving antennas temporarily fail, and cannot collect data for a while. During a seismic survey, a strong explosion may destroy the receiver located in an assumed safe area and information in this area may be lost. Alternatively, several antennas in an array are shutdown to reduce the emitted energy. Such incomplete data, may need to be recollected. However, recollection may be impossible or costly. Hence, processing incomplete data is an alternative solution. In this thesis, low-rank tensor completion algorithms are developed and applications are studied.
In Chapter 1, we introduce the background of the low-rank tensor completion problem. Then, we present recent developments of low-rank tensor completion and its applications. At the end of this chapter, the organization of this thesis is provided.
In Chapter 2, we first present the essential mathematical knowledge for the low-rank tensor completion problem, including the basic tensor algebra and rank function convex optimization. Then, we review the conventional solutions for tensor completion in the literature.
In Chapter 3, we consider low-rank tensor completion in a real-valued and noiseless scenario. A tensor n-mode unfolding truncated nuclear norm is proposed, which is extended from the matrix truncated nuclear norm, to low-rank tensor completion problem. The alternating direction method of multipliers is utilized to solve this optimization problem. Meanwhile, the original two-step solution for the matrix truncated nuclear norm is reduced to one step. The rank information of each tensor unfolding matrix is not required when the intermediate results returned by singular value shrinkage operator is used; thus, the computational complexity of the devised approach is not demanding. Computer simulation results demonstrate the effectiveness of the proposed method.
In Chapter 4, the target estimation problem in a bistatic multiple-input multiple-output radar is addressed via low-rank tensor completion. Our solution consists of jointly computing the direction-of-departure and direction-of-arrival parameters of a sparse target scene, in which only partial data are collected at the front-end during multiple pulse periods. By recasting the data model as a low-rank third-order tensor with missing entries, an accelerated proximal gradient line-search algorithm coupled with rank detection is developed to obtain an accurate rank estimate in a noisy environment with an unknown number of targets. Computer simulation results demonstrate the effectiveness of the proposed method, which outperforms state-of-the-art algorithms that address the same problem.
In Chapter 5, we develop a core tensor thresholding based low-rank tensor completion algorithm. The proposed method utilizes the core tensor thresholding technique to minimize the tensor n-mode rank, which efficiently explores tensor multidimensional structure. Meanwhile, compared with n-mode unfolding based methods, the proposed approach avoids setting the optimal n-mode nuclear norm weighting parameters. The linearized Bregman iteration is employed to achieve a fast convergence. Simulation results verify the effectiveness of the proposed method.
Finally, conclusion of this thesis and future works related to this research are provided in Chapter 6.