矩阵补全模型及其算法研究综述

Survey on Matrix Completion Models and Algorithms

近年来,随着压缩感知技术在信号处理领域的巨大成功,由其衍生而来的矩阵补全技术也日益成为机器学习领域的研究热点,诸多研究者针对矩阵补全问题展开了大量卓有成效的研究.为了更好地把握矩阵补全技术的发展规律,促进矩阵补全理论与工程应用相结合,针对矩阵补全模型及其算法进行了综述.首先,对矩阵补全技术进行溯源,介绍了从压缩感知到矩阵补全的自然演化历程,指出压缩感知理论的发展为矩阵补全理论的形成奠定了基础;其次,从非凸非光滑秩函数松弛的角度将现有矩阵补全模型进行分类,旨在为面向具体应用的矩阵补全问题建模提供思路;然后综述了适用于矩阵补全模型求解的代表性优化算法,其目的在于从本质上理解各种矩阵补全模型优化技巧,从而有利于面向应用问题的矩阵补全新模型求解;最后分析了矩阵补全模型及其算法目前存在的问题,提出了可能的解决思路,并对未来的研究方向进行了展望.

In recent years, with the great success of compressed sensing （CS） in the field of signal processing, matrix completion （MC）, derived from CS, has increasingly become a hot research topic in the field of machine learning. Many researchers have done a lot of fruitful studies on matrix completion problem modeling and their optimization, and constructed relatively complete matrix completiontheory. In order to better grasp the development process of matrix completion, and facilitate the combination of matrix completion theory and engineering applications, this article reviews the existing matrix completion models and their algorithms. First, it introduces the natural evolution process from CS to MC, and illustrates that the development of CS theory has laid the foundation for the formation of MC theory. Second, the article summarizes the existing matrix completion models into the four classes from the perspective of the relaxation of non-convex and non-smooth rank function, aiming to provide reasonable solutions for specific matrix completion applications; Third, in order to understand the inherent optimization techniques and facilitate solving new problem-dependent matrix completion model, the article studies the representative optimization algorithms suitable for various matrix completion models. Finally, article analyzes the existing problems in current matrix completion technology, proposes possible solutions for these problems, and discusses the future work.