基于算子理论的矩阵恢复算法的误差分析

With the rapid development of compressive sensing and sparse representation, the low-rank matrix recovery problem has become a hot topic in the area of machine learning, pattern recognition and computer vision. In compressive sensing and sparse representation, the object we wish to acquire is a vector, while in many applications of practical interest, we often wish to reconstruct an object in the form of a matrix, which is more effective for data acquisition, modeling, processing and analyzing. The research project focus on the low-rank matrix recovery. We explore the organic connection between matrix recovery and statistical learning theory. In the framework of statistical learning theory, we give an all-round analysis on the convergence, stability and the generalization performance of the matrix recovery algorithms under the operator assumptions. The reseach project is aim at improving the convergence rate and the stability of the algorithm, it will promote the basic theoretical achievements extended to the level of technology application. It is the key problem for the practical application of matrix recovery, and is of important theoretical significance and practical value for solving the foregoing problems.

随着压缩传感及稀疏表示理论的迅速发展，矩阵恢复问题逐渐成为机器学习、模式识别及计算机视觉领域中的热点问题。在压缩传感和稀疏表示问题中，待恢复的目标是一个向量，但是在很多实际问题中，待恢复的目标用矩阵来表示的时候对数据的理解、建模、处理和分析更为有效。本项目以低秩矩阵为研究对象，探索矩阵恢复与学习理论的有机联系，从统计学习理论的角度出发，利用算子逼近技术，全面地分析矩阵恢复算法的收敛性、稳定性及推广误差，并将矩阵恢复算法应用于模式识别、图像处理等实际问题。项目以提高算法的收敛速度和稳定性为目标，并将部分基础理论成果推广至应用技术层面。上述问题的解决是将矩阵恢复理论推向实际应用的关键，具有重要的理论意义和实用价值。

本项目从统计学习理论的角度出发，研究低秩矩阵恢复算法的推广误差及稳定性。低秩矩阵恢复是图像处理、信号处理、模式识别等领域中的热点问题，其解的稀疏性和算法稳定性是低秩矩阵恢复理论中两个重要的概念，在设计算法的时候需要同时考虑这两方面的因素。而已有文献证明了稀疏性和稳定性是不可兼得的。因此，在设计算法的时候就必须平衡两者之间的关系。我们研究了低秩矩阵恢复算法的稳定性和推广性能，给出了低秩矩阵恢复算法一致稳定的充分条件，建立了低秩矩阵恢复算法的推广误差与稳定性的联系，并证明了低秩矩阵恢复弹性网elastic-net正则化算法是稳定的。