几类矩阵广义正交约束优化问题的算法、理论及应用

In recent applications from data analysis in data mining and machine learning, several very important problems concerning optimization problems on Stiefel manifold (or the generalized orthogonality constraints set) arise. This project is to study the theory and numerical methods for generalized orthogonality constrained optimization problems and their applications. Based on our previous work along this line, this project aims at 1) establishing relevant and useful results; 2) developing highly efficient and fast algorithms for problems with different structures and different scales. In particular, we will primarily employ the generalized gradient flow method and the corresponding approximate discrete iterative scheme, design the alternating variables method and the symmetric generalized gradient descent method for different expressions of the generalized orthogonality constraints set, study the global convergence result for the hybrid approximated augmented Lagrangian method and the first-order spitting method, propose the accelerated proximal alternating linearized minimization method; and 3) writing Matlab softwares to solve optimization problems according to problem structure and scale, and applying the results into the data analysis fields, such as data mining and machine learning.

矩阵广义正交约束优化问题在数据挖掘、机器学习等数据分析领域有着广泛的应用。本项目以几类矩阵广义正交约束优化问题的理论、算法及应用为研究对象，基于申请人前期关于Stiefel流形上的优化理论与算法的工作，致力于研究这类问题的理论性质，建立适合不同问题结构和规模的高效算法。着重利用广义梯度流构造一阶离散近似算法，针对广义正交约束集的不同表示，设计交替变量法和对称广义梯度下降法。研究近似增广拉格朗日算法框架的全局收敛性，设计加速的邻近交替线性最小化算法。开展数值试验与模拟，在此基础上研制满足不同问题结构和不同规模要求的Matlab软件包。并将结果用于数据挖掘、机器学习等数据分析领域。

本项目致力于研究矩阵（广义）正交约束优化问题的理论及算法，并探索其在大数据分析领域的应用。本项目分别就广义正交约束集上的光滑目标函数及非光滑目标函数展开研究。针对具体问题的结构特征，对于目标函数可微的问题，在流形切空间给出广义内积的定义，基于该定义构造广义梯度; 利用广义梯度构造广义梯度流，并分析其离散迭代近似算法；相关成果发表在Journal of the Franklin Institute. 对于目标函数不可微的问题，分析近似增广Lagrangain算法框架的全局收敛性; 并设计求解近似增广Lagrangian算法子问题的有效加速算法；相关成果发表在Neural Computation. 对于混合高斯脉冲噪声去噪问题，我们构造正交约束下的去噪模型，相关成果发表在 IEEE Transactions on Image Processing.