A Procrustes problem on the Stiefel manifold

Summary. An orthogonal Procrustes problem on the Stiefel manifold is studied, where a matrix Q with orthonormal columns is to be found that minimizes for an matrix A and an matrix B with and . Based on the normal and secular equations and the properties of the Stiefel manifold, necessary conditions for a global minimum, as well as necessary and sufficient conditions for a local minimum, are derived. Received April 7, 1997 / Revised version received April 16, 1998     

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.     

An explicit retraction of symplectic flag manifolds onto complex flag manifolds

We construct an explicit deformation retraction of the manifold of symplectic flags onto the manifold of complex flags. The main tool is the polar decomposition of symplectic matrices. We also give a new definition of symplectic Stiefel manifold and prove that it has the same homotopy type as the complex Stiefel manifold.     

On the Linear Independence of Spikes and Sines

The purpose of this work is to survey what is known about the linear independence of spikes and sines. The paper provides new results for the case where the locations of the spikes and the frequencies of the sines are chosen at random. This problem is equivalent to studying the spectral norm of a random submatrix drawn from the discrete Fourier transform matrix. The proof depends on an extrapolation argument of Bourgain and Tzafriri.     

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.     

Trace ratio optimization with an application to multi-view learning

Mathematical Programming - A trace ratio optimization problem over the Stiefel manifold is investigated from the perspectives of both theory and numerical computations. Necessary conditions in the...     

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.     

Uniform distribution on a Stiefel manifold

We derive various results for the uniform distribution on a Stiefel manifold and propose a test of uniformity.     

多元统计分析中一类矩阵迹函数最小化问题的有效算法

研究来源于多元统计分析中的一类矩阵迹函数最小化问题minc+tr(AX)+∑mj=1tr(BjXCjXT),s.t.XTX = Ip,其中C为常数,A∈Rpxn(n ≥ p),Bj∈Rnxn,Cj∈Rpxp为给定系数矩阵.数值实验表明已有的Majorization算法虽可行,但收敛速度缓慢且精度不高.本文从黎曼流形的角...     

Two adaptive scaled gradient projection methods for Stiefel manifold constrained optimization

Numerical Algorithms - This article is concerned with the problem of minimizing a smooth function over the Stiefel manifold. In order to address this problem, we introduce two adaptive scaled...     

H2 optimal model order reduction on the Stiefel manifold for the MIMO discrete system by the cross Gramian

ABSTRACT

In this paper, the H₂ optimal model order reduction method for the large-scale multiple-input multiple-output (MIMO) discrete system is investigated. First, the MIMO discrete system is resolved into a number of single-input single-output (SISO) subsystems, and the H₂ norm of the original MIMO discrete system is expressed by the cross Gramian of each subsystem. Then, the retraction and the vector transport on the Stiefel manifold are introduced, and the geometric conjugate gradient model order reduction method is proposed. The reduced system of the original MIMO discrete system is generated by using the proposed method. Finally, two numerical examples show the efficiency of the proposed method.     

Probability Theory of Random Polygons from the Quaternionic Viewpoint

We build a new probability measure on closed space and plane polygons. The key construction is a map, given by Hausmann and Knutson, using the Hopf map on quaternions from the complex Stiefel manifold of 2‐frames in n‐space to the space of closed n‐gons in 3‐space of total length 2. Our probability measure on polygon space is defined by pushing forward Haar measure on the Stiefel manifold by this map. A similar construction yields a probability measure on plane polygons that comes from a real Stiefel manifold. The edgelengths of polygons sampled according to our measures obey beta distributions. This makes our polygon measures different from those usually studied, which have Gaussian or fixed edgelengths. One advantage of our measures is that we can explicitly compute expectations and moments for chord lengths and radii of gyration. Another is that direct sampling according to our measures is fast (linear in the number of edges) and easy to code. Some of our methods will be of independent interest in studying other probability measures on polygon spaces. We define an edge set ensemble (ESE) to be the set of polygons created by rearranging a given set of n edges. A key theorem gives a formula for the average over an ESE of the squared lengths of chords skipping k vertices in terms of k, n, and the edgelengths of the ensemble. This allows one to easily compute expected values of squared chord lengths and radii of gyration for any probability measure on polygon space invariant under rearrangements of edges. © 2014 Wiley Periodicals, Inc.     

Numerical study of learning algorithms on Stiefel manifold

Convex optimization methods are used for many machine learning models such as support vector machine. However, the requirement of a convex formulation can place limitations on machine learning models. In recent years, a number of machine learning methods not requiring convexity have emerged. In this paper, we study non-convex optimization problems on the Stiefel manifold in which the feasible set consists of a set of rectangular matrices with orthonormal column vectors. We present examples of non-convex optimization problems in machine learning and apply three nonlinear optimization methods for finding a local optimal solution; geometric gradient descent method, augmented Lagrangian method of multipliers, and alternating direction method of multipliers. Although the geometric gradient method is often used to solve non-convex optimization problems on the Stiefel manifold, we show that the alternating direction method of multipliers generally produces higher quality numerical solutions within a reasonable computation time.     

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.     

Concentrated matrix Langevin distributions

This paper concerns the matrix Langevin distributions, exponential-type distributions defined on the two manifolds of our interest, the Stiefel manifold V_k,m and the manifold P_k,m−k of m×m orthogonal projection matrices idempotent of rank k which is equivalent to the Grassmann manifold G_k,m−k. Asymptotic theorems are derived when the concentration parameters of the distributions are large. We investigate the asymptotic behavior of distributions of some (matrix) statistics constructed based on the sample mean matrices in connection with testing hypotheses of the orientation parameters, and obtain asymptotic results in the estimation of large concentration parameters and in the classification of the matrix Langevin distributions.     

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.     

线性流形上广义中心反对称矩阵的最佳逼近

本讨论在线性流形上广义反对称矩阵的最佳逼近，给出了若干有意义的结果。     

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.     

Sectional curvatures in nonlinear optimization

The aim of the paper is to show how to explicitly express the function of sectional curvature with the first and second derivatives of the problem’s functions in the case of submanifolds determined by equality constraints in the n-dimensional Euclidean space endowed with the induced Riemannian metric, which is followed by the formulation of the minimization problem of sectional curvature at an arbitrary point of the given submanifold as a global minimization one on a Stiefel manifold. Based on the results, the sectional curvatures of Stiefel manifolds are analysed and the maximal and minimal sectional curvatures on an ellipsoid are determined. This research was supported in part by the Hungarian Scientific Research Fund, Grant No. OTKA-T043276 and OTKA-K60480.