A Study on Weighted and Structured Low Rank Matrix Optimization Problems

Abstract

This thesis focuses on the weighted and structured low rank approximation problem (wSLRA). This problem arises from a wide range of applications such as signal recovery, image processing and matrix learning. Due to the non-convexity of low rank matrix set, this problem is NP-hard and difficult to tackle. In this thesis we firstly focus on weighted low rank Hankel matrix optimization problem, which has become one of the main approaches to the signal extraction from noisy series or signals of finite rank by selecting the suitable weight matrix.
Two guiding principles for developing an approach are (i) the Hankel matrix optimization should be computationally tractable, and (ii) the objective in the optimization should be a close approximation to the original weighted least-squares. In this thesis we firstly introduce an approach that satisfies (i) and (ii) called Sequential Majorization Method (SMM). The framework of majorization method introduces guaranteed convergence by successfully solving the subproblem of each iterate, which ensures the sandwich inequality. At the same time, the latest gradient information is used when solving the to approximate weight norm are not mathematically equivalent, and the alternating projection method is used when optimizing the subproblem which has no convergence
guarantees.
We further propose a new scheme as PenalisedMethod of Alternating Projection (pMAP). The proposed method inherits the favourable local properties of MAP and has the same computational complexity. Moreover, it is capable of handling a general weight matrix,
is globally convergent, and enjoys local linear convergence rate provided that the cutting o↵ singular values are significantly smaller than the kept ones. Furthermore, the new method also applies to complex data. Extensive numerical experiments demonstrate the efficiency of the proposed method against several popular variants of MAP.
This pMAP scheme is further extended to solve a wider rang of structured low rank matrix optimization problems such as robust matrix completion and robust principal component analysis with small noise. We use these two examples to demonstrate the approach to extend pMAP framework while keeping its advantages in dealing with low rank matrix approximation problems including computing efficiency and convergence results. Numerical experiments are conducted for both examples to illustrate the competitiveness of pMAP comparing with some state-of-the-art solvers subproblem of SMM, leading to more accurate approximations to the objective. However, the SMM scheme still has some drawbacks. The (q, p)-norm introduced by SMM

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ORCID for Hou-Duo Qi: orcid.org/0000-0003-3481-4814

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