低秩矩阵复原的Schatten-q(0<q<1)正则化理论与算法研究

Low-rank matrix recovery, is a technique for reconstructing a matrix using a small number of linear or nonlinear measurements that the matrix's rank is low. Matrix recovery has a strong background in application. However, the effectiveness of its application is strongly depends on the application skills (for example, recovery theory, algorithm design, measurement operator and the choice of the number of samples, etc.). This situation call for a thorough understanding on the nature of matrix recovery, and hope a breakthrough of the core foundation of matrix recovery. In this project, we will make a systematic and deep investigation into the recoverable capability of low-rank matrix, the following is the three major aspects of research: (1). Using the theory on the geometric structure of Banach space, the new measure method will be investigated, and the equivalent theory between Shatten-q and Shatten-0 will be studied. (2). Based on the Shatten-q regularization, the upper bound, the lower bound and the essential bound will be developed, and the relationship among the recoverable capability, the measurement operator, and the sample number will be clarified. (3).Using the above theory, the appropriate algorithm will be established, and applied to recovery of face image, assessment and forecast of online recommendation system, computer vision, etc. The implementation of this project will lay mathematical foundations on the theory and applications of low-rank matrix recovery,and promote the further development of matrix recovery.

低秩矩阵复原是当原始矩阵呈现低秩特征时，通过测量算子的某种线性或非线性运算后的少量元素，来精确恢复原始矩阵的新理论，具有很强的应用背景。然而，其应用的有效性强烈地依赖应用的技巧性（如复原的理论基础、算法设计、测量算子及采样数目等）。这一现状呼唤对矩阵复原本质性态的透彻理解，呼唤对矩阵复原核心基础的突破。本项目围绕此目标将对低秩矩阵的复原能力展开深入系统地研究，主要有：（1）利用Banach空间几何结构理论，探讨测量算子新的度量方式；研究Schatten-0与Schatten-q范数的等价性理论；（2）基于Schatten-q正则化，研究复原速度的上、下界估计和本质复原阶估计；澄清复原能力与测量算子、采样数目之间的内在联系；（3）建立相应的算法，应用到人脸图像复原、在线推荐系统评价预测、计算机视觉等问题。此项目实施将为低秩矩阵复原的理论和应用奠定数学基础，对矩阵复原的进一步发展具有重要意义。

矩阵重构是当一个矩阵可压缩或可稀疏表示时，通过测量算子的某种线性或非线性运算后的元素，来精确地地重构出原始矩阵的新理论，具有很强的应用背景。 然而，其应用的有效性强烈地依赖应用的技巧性（如可重构的理论基础、算法设计、测量算子及采样数目的选择等）。这一现状呼唤对矩阵重构本质性态的透彻理解，呼唤对矩阵重构核心基础问题的突破。本项目围绕这一目标对低秩（低阶、高阶）矩阵的重构能力展开深入系统地研究，主要进行三方面研究：（1）利用Banach空间几何结构理论，探讨测量算子新的可度量方式；（2）基于Lq正则化，研究了低阶矩阵（向量）和高阶矩阵（张量）情形下，低秩矩阵重构速度的估计；澄清重构能力与测量算子、采样数目之间的内在联系；（3）发展并建立了相应的算法，应用到图像的恢复与重建、背景分离等问题。