低秩矩阵恢复理论与算法研究

Low-rank matrix recovery, derived from the recently popular theory of compressive sensing and sparse representation, is a hot topic in the area of image processing, signal processing, pattern recognition and machine learning. In compressive sensing and sparse representation, the objects we wish to acquire are vectors, 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.

低秩矩阵恢复衍生于近几年非常流行的压缩传感及稀疏表示理论，是图像处理、信号处理、模式识别及优化领域中的研究热点。在压缩传感和稀疏表示问题中，待恢复的目标都是向量，而在很多实际应用中，待恢复的目标用矩阵表示的时候对问题的理解、建模、处理和分析更为有效。本项目以低秩矩阵为研究对象，围绕低秩矩阵恢复问题中模型的建立、算法的设计、误差分析这三个核心问题展开研究。建立低秩矩阵恢复的最优化模型，设计算法对模型进行求解。针对现有算法耗时长、不稳定的问题，设计低秩矩阵恢复在线学习算法，对算法的收敛性、稳定性及推广性进行分析。本项目以提高低秩矩阵恢复算法的收敛速度和稳定性为目标，将部分理论成果推广至应用技术层面。上述问题的解决是将低秩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。

随着压缩感知及稀疏表示问题的流行，低秩矩阵恢复问题成为机器学习、模式识别及计算机视觉领域的一大热点。本项目结合统计学习理论、稀疏表示及低秩矩阵恢复中的方法和技巧，研究了原子范数正则化模型，构造了基于多项式核的原子范数正则化算法，并且基于统计学习理论中误差分析的方法对原子范数正则化算法的推广性能进行了分析。此外，基于稀疏表示和正则化方法提出一种新的高光谱图像分类算法，近邻正则化联合稀疏表示算法，并将其应用到高光谱图像分类中，取得了很好的分类效果。进一步，对于固定设计高斯回归问题，我们提出了一种惩罚经验松弛贪婪算法，并且通过建立相应的oracle不等式分析了算法的有效性。