正交约束优化问题的一阶算法

带有正交约束的矩阵优化问题在材料计算、统计及数据分析等领域中有着广泛的应用.由于正交约束的可行域是Stiefel流形,一直以来流形上的优化方法是求解这一问题的主要方法.近年来,随着实际应用问题所要求的变量规模的扩大,传统的流形优化方法在计算上的劣势显现出来,而一些迭代简单、收敛快的新算法逐渐被提出.通过收缩方法、非收缩可行方法、不可行方法三个类别分别来介绍求解带有正交约束的矩阵优化问题的最新算法.通过分析这些方法的主要特性,以及应用问题的要求,对这类问题算法设计的研究进行了展望.     

列正交约束下广义Sylvester方程极小化问题的有效算法

研究列正交约束下广义Sylvester方程极小化问题的有效算法.基于Stiefel流形的几何性质和欧氏空间中的MPRP共轭梯度法,构造一类黎曼MPRP共轭梯度迭代求解算法,给出算法全局收敛性.该迭代格式得到的搜索方向总能保证该目标函数下降.数值实验和数值比较验证所提出算法对于问题模型是高效可行的.     

复矩阵截断奇异值分解的一类混合算法

截断奇异值分解是一类非常重要的矩阵分解,其在病态模型问题分析等领域有广泛的应用.该文主要研究复矩阵截断奇异值分解的有效算法,将问题转化为复Stiefel乘积流形上的黎曼优化问题,进而设计基于乘积流形的黎曼混合牛顿法求解.为有效求解黎曼牛顿方程,从降低系统维数和简化计算入手,通过克罗内克积和复矩阵拉直算子将其转化为易于求解的标准实对称线性方程组.数值实验和数值比较验证该文所提算法针对复矩阵截断奇异值分解问题是高效可行的.     

透镜空间L~n(4)上的向量丛及相关的对合

本文研究了透镜空间Ln(4)的上同调群生成元的运算性质,利用这些生成元,并借助于KO-理论计算出了透镜空间Ln(4)上任意向量丛的全Stiefel—Whitney示性类.更进一步地,给出了不动点集为透镜空间Ln(4)的带对合的流形的协边分类.     

Stiefel流形上的黎曼共轭梯度法CSCD

1引言Stiefel流形上的优化问题一般形式可以表示为:min x∈S_(n,p) f(X)(1.1)其中目标函数f:R^(n×p)→R为连续可微函数,S_(n,p)表示Stiefel流形,即S_(n,p)={X∈R^(n×p):X^(T)X=Ip,p相似文献   

复Stiefel流形的商空间

本文研究了复Stiefel流形关于群S1的商空间的伦型,并且计算了该空间的上同调群.通过纤维化,为这些商空间的上同调找到一组典则的生成元.再利用推广的吴公式,讨论了这些生成元在Sqi下的行为.最后,作为应用,本文对S.Gitler和D.Handel的结果作了部分改进.     

黎曼流形上次梯度算法及其收敛性的研究

主要研究了黎曼流形M上一类最优化问题,并给出了解决该问题的一种ε次梯度算法.并在流形M是一个完备的且具有非负截面曲率的黎曼流形时,证明了算法得出的无限迭代点列的收敛性.     

几种典型数据类型的谱聚类方法研究

主要针对几种典型数据的多流形结构分析问题进行了研究.综合分析多种谱聚类算法优缺点,以谱多流形聚类算法为主线,结合实验结果对多种谱聚类算法进行了分析,最后针对数据空间密度不均匀的情况对谱多流形聚类算法进行了一定的改进,提出了一种基于自适应近邻值的谱多流形聚类算法,并通过实验证明其达到了混合多流形聚类的目的.     

可拓展概率逼近中一类矩阵优化问题的有效算法

研究来源于复杂系统离散逼近中的一类可拓展概率逼近模型,欧氏空间中该问题模型可重塑为一类由线性流形和斜流形组成的乘积流形约束矩阵优化问题.结合乘积流形的几何性质,基于Zhang-Hager技术拓展,本文设计一类适用于问题模型的黎曼非线性共轭梯度法,并给出算法全局收敛性分析.数值实验验证所提算法对于问题模型求解是高效可行的,且与其它黎曼梯度类算法及黎曼优化工具箱中已有的黎曼梯度类算法和二阶算法相比在迭代效率上有一定优势.     

广义特征值极小扰动问题的一类黎曼共轭梯度法

研究含参数$l$非方矩阵对广义特征值极小扰动问题所导出的一类复乘积流形约束矩阵最小二乘问题.与已有工作不同,本文直接针对复问题模型,结合复乘积流形的几何性质和欧式空间上的改进Fletcher-Reeves共轭梯度法,设计一类适用于问题模型的黎曼非线性共轭梯度求解算法,并给出全局收敛性分析.数值实验和数值比较表明该算法比参数$l=1$的已有算法收敛速度更快,与参数$l=n$的已有算法能得到相同精度的解.与部分其它流形优化相比与已有的黎曼Dai非线性共轭梯度法具有相当的迭代效率,与黎曼二阶算法相比单步迭代成本较低、总体迭代时间较少,与部分非流形优化算法相比在迭代效率上有明显优势.     

Some Global Optimization Problems on Stiefel Manifolds

Optimization on Stiefel manifolds was discussed by Rapcsák in earlier papers, and some global optimization methods were considered and tested on Stiefel manifolds. In the paper, test functions are given with known global optimum points and their optimal function values. A restriction, which leads to a discretization of the problem is suggested, which results in a problem equivalent to the well-known assignment problem.     

Nonmonotone algorithm for minimization on closed sets with applications to minimization on Stiefel manifolds

A nonmonotone Levenberg–Marquardt-based algorithm is proposed for minimization problems on closed domains. By preserving the feasible set’s geometry throughout the process, the method generates a feasible sequence converging to a stationary point independently of the initial guess. As an application, a specific algorithm is derived for minimization on Stiefel manifolds and numerical results involving a weighted orthogonal Procrustes problem are reported.     

State space models on special manifolds

This paper concerns modeling time series observations in state space forms considered on the Stiefel and Grassmann manifolds. We develop a state space model relating the time series observations to a sequence of unobserved state or parameter matrices assuming the matrix Langevin noise processes on the Stiefel manifolds. We show a Bayes method for estimating the state matrices by the posterior modes. We consider a further extended state space model where two sequences of unobserved state matrices are involved. A simple state space model on the Grassmann manifolds with matrix Langevin noise processes is also investigated.     

The Mitchell-Richter filtration of loops on Stiefel manifolds stably splits

We prove that the Mitchell-Richter filtration of the space of loops on complex Stiefel manifolds stably splits. The result is obtained as a special case of a more general splitting theorem. Another special case is H. Miller's splitting of Stiefel manifolds. The proof uses the theory of orthogonal calculus developed by M. Weiss. The argument is inspired by an old argument of Goodwillie for a different, but closely related, general splitting result.

     

A matrix-free implementation of Riemannian Newton’s method on the Stiefel manifold

Newton’s method for unconstrained optimization problems on the Euclidean space can be generalized to that on Riemannian manifolds. The truncated singular value problem is one particular problem defined on the product of two Stiefel manifolds, and an algorithm of the Riemannian Newton’s method for this problem has been designed. However, this algorithm is not easy to implement in its original form because the Newton equation is expressed by a system of matrix equations which is difficult to solve directly. In the present paper, we propose an effective implementation of the Newton algorithm. A matrix-free Krylov subspace method is used to solve a symmetric linear system into which the Newton equation is rewritten. The presented approach can be used on other problems as well. Numerical experiments demonstrate that the proposed method is effective for the above optimization problem.     

Morse�CBott functions and the Lusternik�CSchnirelmann category

Lusternik–Schnirelmann category of a manifold gives a lower bound of the number of critical points of a differentiable map on it. The purpose of this paper is to show how to construct cone-decompositions of manifolds by using functions of class C ¹ and their gradient flows, where cone-decompositions are used to give an upper bound for the Lusternik–Schnirelmann category which is a homotopy invariant of a topological space. In particular, the Morse–Bott functions on the Stiefel manifolds considered by Frankel (1965) are effectively used to construct the conedecompositions of Stiefel manifolds and symmetric Riemannian spaces to determine their Lusternik–Schnirelmann categories.     

Properly discontinuous actions on Hilbert manifolds

In this article we study properly discontinuous actions on Hilbert manifolds giving new examples of complete Hilbert manifolds with nonnegative, respectively nonpositive, sectional curvature with infinite fundamental group. We also get examples of complete infinite dimensional Kähler manifolds with positive holomorphic sectional curvature and infinite fundamental group in contrastwith the finite dimensional case and we classify abelian groups acting linearly, isometrically and properly discontinuously on Stiefel manifolds. Finally, we classify homogeneous Hilbert manifolds with constant sectional curvature.     

L-S category of quaternionic Stiefel manifolds

By calculating certain generalized cohomology theory, lower bounds for the L-S category of quaternionic Stiefel manifolds are given.     

Parallelizability of complex projective Stiefel manifolds

The question of parallelizability of the complex projective Stiefel manifolds is settled.

     

Closed-form Geodesics and Optimization for Riemannian Logarithms of Stiefel and Flag Manifolds

Journal of Optimization Theory and Applications - We provide two closed-form geodesic formulas for a family of metrics on Stiefel manifolds recently introduced by Hüper, Markina and Silva...