低秩矩阵复原的Schatten-q(0

项目名称： 低秩矩阵复原的Schatten-q(0

项目编号： No.61273020

项目类型： 面上项目

立项/批准年度： 2013

项目学科： 自动化技术、计算机技术

项目作者： 王建军

作者单位： 西南大学

项目金额： 58万元

中文摘要： 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。

中文关键词： 矩阵修补；压缩感知；张量恢复；低秩；

英文摘要： 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 recommenda

英文关键词： Matrix completion；Compressd sensing；Tensor recovery；low rank；