基于约束等距条件的噪音低秩矩阵恢复算法研究

With the development of information technology, high-dimensonal data processing has become one of the most important research problems in the field of information theory, biologic science, engineering and medicine, etc. Particllarly, when high-dimensonal data is represented as a matrix, low rank matrix recovery, which has developed rapidly in applied mathematics in recent years, has brought new solution method for high-dimensonal data processing. Low rank matrix recovery predicts that the unknown low-rank matrices can be recovered from fewer number of linear measurements with high probabilities. Low rank matrix recovery has various applications in the field of image processing, online recommendation, etc. This new theory has many unsolved problems.. This project will study some recovery algorithms in noisy low rank matrix recovery problems, including convex, nonconvex Schatten-p norm minimization method and iteratively reweighted least square algorithm. Based on the restricted isometry property of linear measurement map, we will first give the robust recovery analysis adapting to the measurement noise of convex and nonconvex Schatten-p norm minimization method. Then we will give convergence, stability and robust analysis of iteratively reweighted least square algorithm for noisy low rank matrix recovery. We hope our results will provide theoretical support for the applications of low rank matrix recovery.

随着信息技术的不断发展，高维数据处理问题已经成为信息科学、生命科学、工程以及医学等领域的重要研究问题之一。特别地，当高维数据表示为矩阵时，近年来应用数学领域迅速发展的低秩矩阵恢复理论为高维数据处理问题提供了新的解决方法。低秩矩阵恢复理论表明可以从远少于矩阵元素个数的线性观测向量中以高概率精确恢复未知低秩矩阵。低秩矩阵恢复理论在图像处理及在线推荐系统等领域有着广泛的应用。这一新兴的学科还有许多待解决的理论问题。. 本项目将研究噪音测量下低秩矩阵恢复问题的几类重构算法，重点研究凸和非凸的Schatten-p范数最小化方法和迭代加权最小二乘算法。基于测量映射所满足的约束等距条件，本项目将给出凸和非凸的Schatten-p范数最小化方法自适应于噪音的鲁棒性恢复结果；给出迭代加权最小二乘算法在噪音测量下的收敛性、稳定性和鲁棒性分析，从而为低秩矩阵恢复的实际应用提供理论支撑。

低秩矩阵恢复理论及其应用是目前应用数学领域一个非常具有发展前景的研究方向。低秩矩阵恢复理论考虑通过远少于未知矩阵元素个数的线性观测向量高概率精确恢复低秩矩阵。本项目预定研究内容大致可以分为两个方面：. 1.迭代加权最小二乘算法的收敛性及收敛速率分析；. 2.低秩矩阵恢复中的Schatten-p范数最小化理论。. 项目组成员按原定计划顺利开展了研究，并且取得了一批较好的研究成果，给出了迭代加权最小二乘算法在噪音测量下的收敛性及稳定性分析，证明了该算法具有线性收敛速率；通过研究压缩感知中lp最小化方法的约束等距常数界进而给出Schatten-p范数最小化方法精确及鲁棒地恢复低秩矩阵的最优矩阵约束等距常数界估计，该估计解决了T.Cai及A.Zhang论文中关于约束等距常数的一个猜测。. 本项目组成员共发表与项目研究内容相关的学术论文（SCI源刊物）2篇，另有被SCI源刊物接收录用的论文1篇。可以说已较为圆满地完成了项目的预定研究目标。