Quadratic programs over the Stiefel manifold

We characterize the optimal solution of a quadratic program over the Stiefel manifold with an objective function in trace formulation. The result is applied to relaxations of HQAP and MTLS. Finally, we show that strong duality holds for the Lagrangian dual, provided some redundant constraints are added to the primal program.     

Stiefel流形上的梯度下降法

基于Stiefel流形上算法的几何框架,本文提出了Stiefel流形上的梯度下降法.理论上给出了算法收敛性定理.三个数值仿真算例表明算法是有效的,与其他方法相比具有更快的收敛速度.     

Non-neutrality of the Stiefel manifolds Vn,k II

The Stiefel manifolds V_2^m−1,k are shown to be non-neutral for m5, 2^m−1+2k=2ℓ<2^m−2.     

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.

     

Distributions of orientations on Stiefel manifolds

The Riemann space whose elements are m × k (m k) matrices X, i.e., orientations, such that X′X = I_k is called the Stiefel manifold V_k,m. The matrix Langevin (or von Mises-Fisher) and matrix Bingham distributions have been suggested as distributions on V_k,m. In this paper, we present some distributional results on V_k,m. Two kinds of decomposition are given of the differential form for the invariant measure on V_k,m, and they are utilized to derive distributions on the component Stiefel manifolds and subspaces of V_k,m for the above-mentioned two distributions. The singular value decomposition of the sum of a random sample from the matrix Langevin distribution gives the maximum likelihood estimators of the population orientations and modal orientation. We derive sampling distributions of matrix statistics including these sample estimators. Furthermore, representations in terms of the Hankel transform and multi-sample distribution theory are briefly discussed.     

Runge-Kutta Type Methods Based on Geodesics for Systems of ODEs on the Stiefel Manifold

In this paper we propose numerical methods for solving ODEs on the Stiefel manifold based on the use of the embedded geodesics. Numerical tests are also provided in order to show the features of our methods.This revised version was published online in October 2005 with corrections to the Cover Date.     

A Low Complexity Lie Group Method on the Stiefel Manifold

A low complexity Lie group method for numerical integration of ordinary differential equations on the orthogonal Stiefel manifold is presented. Based on the quotient space representation of the Stiefel manifold we provide a representation of the tangent space suitable for Lie group methods. According to this representation a special type of generalized polar coordinates (GPC) is defined and used as a coordinate map. The GPC maps prove to adapt well to the Stiefel manifold. For the n×k matrix representation of the Stiefel manifold the arithmetic complexity of the method presented is of order nk ², and for nk this leads to huge savings in computation time compared to ordinary Lie group methods. Numerical experiments compare the method to a standard Lie group method using the matrix exponential, and conclude that on the examples presented, the methods perform equally on both accuracy and maintaining orthogonality.     

Hypercomplex structures on Stiefel manifolds

This paper describes a family of hypercomplex structures {% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf% gDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWFqessaaa!4076!\[\mathcal{I}\]a(p)}a=1,2,3 depending on n real non-zero parameters p = (p ₁,...,p n) on the Stiefel manifold of complex 2-planes in n for all n > 2. Generally, these hypercomplex structures are inhomogenous with the exception of the case when all the p i's are equal. We also determine the Lie algebra of infinitesimal hypercomplex automorphisms for each structure. Furthermore, we solve the equivalence problem for the hypercomplex structures in the case that the components of p are pairwise commensurable. Finally, some of these examples admit discrete hypercomplex quotients whose topology we also analyze.During the preparation of this work all three authors were supported by NSF grants.     

Density Estimation on the Stiefel Manifold

This paper develops the theory of density estimation on the Stiefel manifoldV_{k, m}, whereV_{k, m}is represented by the set ofm×kmatricesXsuch thatX′X=I_k, thek×kidentity matrix. The density estimation by the method of kernels is considered, proposing two classes of kernel density estimators with small smoothing parameter matrices and for kernel functions of matrix argument. Asymptotic behavior of various statistical measures of the kernel density estimators is investigated for small smoothing parameter matrix and/or for large sample size. Some decompositions of the Stiefel manifoldV_{k, m}play useful roles in the investigation, and the general discussion is applied and examined for a special kernel function. Alternative methods of density estimation are suggested, using decompositions ofV_{k, m}.     

Compact Kähler manifolds with elliptic homotopy type

Simply connected compact Kähler manifolds of dimension up to three with elliptic homotopy type are characterized in terms of their Hodge diamonds. For surfaces there are only two possibilities, namely h1,1?2 with hp,q=0 for p≠q. For threefolds, there are three possibilities, namely h1,1?3 with hp,q=0 for p≠q. This characterization in terms of the Hodge diamonds is applied to explicitly classify the simply connected Kähler surfaces and Fano threefolds with elliptic homotopy type.     

Maximization of the sum of the trace ratio on the Stiefel manifold,I: Theory

We are concerned with the maximization of tr(VTAV)/tr(VT BV)+ tr(VT CV)over the Stiefel manifold {V ∈ Rm×| V T V = It}(t m), where B is a given symmetric and positive definite matrix, A and C are symmetric matrices, and tr() is the trace of a square matrix. This is a subspace version of the maximization problem studied in Zhang(2013), which arises from real-world applications in, for example,the downlink of a multi-user MIMO system and the sparse Fisher discriminant analysis in pattern recognition.We establish necessary conditions for both the local and global maximizers and connect the problem with a nonlinear extreme eigenvalue problem. The necessary condition for the global maximizers offers deep insights into the problem, on the one hand, and, on the other hand, naturally leads to a self-consistent-field(SCF)iteration to be presented and analyzed in detail in Part II of this paper.     

Homotopy connectedness theorems for submanifolds of Sasakian manifolds

The homotopy connectedness theorem for invariant immersions in Sasakian manifolds with nonnegative transversal q-bisectional curvature is proved. Some Barth-Lefschetz type theorems for minimal submanifolds and (k, ?)-saddle submanifolds in Sasakian manifolds with positive transversal q-Ricci curvature are proved by using the weak (?-)asymptotic index. As a corollary, the Frankel type theorem is proved.     

Homotopy minimal period self-maps on flat manifolds

This paper studies the homotopical minimal period set for self-maps on flat manifolds. We introduce a new invariant: the density of the homotopical minimal period set in the natural numbers. Some conditions on self-maps and flat manifolds are obtained so that the corresponding density is positive.     

Inductive Detection for Homotopy Equivalences of Manifolds

Let f:M N be a homotopy equivalence of CAT manifolds M and N (CAT := PL, TOP or DIFF) with finite fundamental groups. Each subgroup H 1(M) determines a homotopy equivalence fH:MH NH of the corresponding covering spaces. Suppose now that for each subgroup H in some particular class C (for example: elementary, hyperelementary or solvable) fH is homotopic to a CAT isomorphism. The general problem studied in this paper can be formulated as follows: If each map fH as above is homotopic to a CAT isomorphism, under what additional conditions on M, C and CAT is f itself (or f × idR) homotopic (properly homotopic) to a CATisomorphism?     

Riemannian Newton-type methods for joint diagonalization on the Stiefel manifold with application to independent component analysis

The joint approximate diagonalization of non-commuting symmetric matrices is an important process in independent component analysis. This problem can be formulated as an optimization problem on the Stiefel manifold that can be solved using Riemannian optimization techniques. Among the available optimization techniques, this study utilizes the Riemannian Newton’s method for the joint diagonalization problem on the Stiefel manifold, which has quadratic convergence. In particular, the resultant Newton’s equation can be effectively solved by means of the Kronecker product and the vec and veck operators, which reduce the dimension of the equation to that of the Stiefel manifold. Numerical experiments are performed to show that the proposed method improves the accuracy of the approximate solution to this problem. The proposed method is also applied to independent component analysis for the image separation problem. The proposed Newton method further leads to a novel and fast Riemannian trust-region Newton method for the joint diagonalization problem.     

Lyapunov exponents and control sets near singular points

We consider control affine systems of the form

     

Collapses, products and LC manifolds

The quantum physicists Durhuus and Jonsson (1995) [9] introduced the class of “locally constructible” (LC) triangulated manifolds and showed that all the LC 2- and 3-manifolds are spheres. We show here that for each d>3 some LC d-manifolds are not spheres. We prove this result by studying how to collapse products of manifolds with one facet removed.     

一类具有周期变指数和凹凸非线性项椭圆型方程解的多重性

研究一类具有周期变指数和凹凸非线性项的椭圆边值问题,借Ekeland分原理和Nehari流形等理论和方法得到解的多重性.     

On a sub‐Stiefel Procrustes problem arising in computer vision

A sub‐Stiefel matrix is a matrix that results from deleting simultaneously the last row and the last column of an orthogonal matrix. In this paper, we consider a Procrustes problem on the set of sub‐Stiefel matrices of order n. For n = 2, this problem has arisen in computer vision to solve the surface unfolding problem considered by R. Fereirra, J. Xavier and J. Costeira. An iterative algorithm for computing the solution of the sub‐Stiefel Procrustes problem for an arbitrary n is proposed, and some numerical experiments are carried out to illustrate its performance. For these purposes, we investigate the properties of sub‐Stiefel matrices. In particular, we derive two necessary and sufficient conditions for a matrix to be sub‐Stiefel. We also relate the sub‐Stiefel Procrustes problem with the Stiefel Procrustes problem and compare it with the orthogonal Procrustes problem. Copyright © 2015 John Wiley & Sons, Ltd.