两类特殊非负矩阵分解问题的黎曼优化方法研究

In this project, we are mainly concerned with two special kinds of nonnegative matrix factorizations: the semi-orthogonal nonnegative matrix factorization and the nonnegative matrix factorization based on the generalized Kullback-Leibler divergence. Most existing researches for these two kinds of problems devise numerical methods from constrained Euclidean optimization viewpoint, while we wish to construct some stable and efficient Riemannain optimization algorithms from Riemannian optimization viewpoint. Based on the special structures of the original problems, we can reformulate the first kind of problem into equivalent convex constrained Riemannian optimization problems and complimentary problems with Riemannian contraints, and reformulate the second kind of problem into equivalent unconstrained Riemanninan optimization problems. We will try to construct some Riemannian optimization methods to solve these equivalent optimization problems and complimentary problems defined on product Riemannian manifolds from the viewpoints of simultaneous minimization and alternating minimization, which mainly includes algorithm constructions for convex constrained Riemannian convex and nonconvex optimization problems, linear and nonlinear complimentary problems with Riemannian constraints, and unconstrained Riemannian convex and nonconvex optimization problems. We will also investigate the convergence and stability properties of these algorithms, and extent these methods to solve some important realistic application problems.

在本项目中，我们主要考虑两类特殊的非负矩阵分解问题：半正交非负矩阵分解问题以及基于广义Kullback-Leibler散度的非负矩阵分解问题。现有的关于这两类问题的研究多数是从欧式空间约束优化的角度设计算法，我们则希望从黎曼优化的角度设计稳定而有效的黎曼优化算法来求解这两类特殊问题。通过分析问题本身的特性，我们将第一类问题等价转化为凸约束黎曼优化问题以及具有黎曼约束的互补问题，将第二类问题等价转为为无约束黎曼优化问题。我们将分别从同时优化和交替优化的角度出发去设计黎曼优化算法求解定义在黎曼乘积流形上的等价优化和互补问题，其中主要包括凸约束黎曼凸和非凸优化问题、具有黎曼约束的线性和非线性互补问题以及无约束的黎曼凸和非凸优化问题的算法设计。我们将分析算法的收敛性和稳定性，并将算法推广应用于求解一些实际应用问题。

在本项目中，我们主要研究两类特殊的非负矩阵分解问题以及相关的黎曼优化问题。原有的算法主要从欧式空间约束优化的角度构造数值算法，而我们则希望从黎曼优化的角度设计稳定且有效的算法。针对黎曼流形上的无约束优化问题、欠定方程求解、最小二乘问题、切向量场零点求解等问题，我们设计了黎曼修正Fletcher-Reeves非线性共轭梯度法、非精确黎曼牛顿法、预处理黎曼高斯牛顿法、非单调黎曼derivative-free PRP类型算法以及黎曼谱共轭梯度法，这些算法可稳定有效地应用于求解非负矩阵分解问题、给定部分特征对的随机矩阵特征值反问题、非负矩阵特征值反问题、参数化最小二乘特征值反问题等问题的求解。此外，我们还给出了某些Hermitian不定线性系统的MINRES迭代算法的收敛界。