In this paper,we present a novel penalty model called ExPen for optimization over the Stiefel manifold.Different from existing penalty functions for orthogonality constraints,ExPen adopts a smooth penalty function wit...In this paper,we present a novel penalty model called ExPen for optimization over the Stiefel manifold.Different from existing penalty functions for orthogonality constraints,ExPen adopts a smooth penalty function without using any first-order derivative of the objective function.We show that all the first-order stationary points of ExPen with a sufficiently large penalty parameter are either feasible,namely,are the first-order stationary points of the original optimization problem,or far from the Stiefel manifold.Besides,the original problem and ExPen share the same second-order stationary points.Remarkably,the exact gradient and Hessian of ExPen are easy to compute.As a consequence,abundant algorithm resources in unconstrained optimization can be applied straightforwardly to solve ExPen.展开更多
We consider the unbalanced Procrustes problem with an orthonormal constraint on solutions： given matrices A ∈ R^n×n and B ∈ R^n×k, n 〉 k, minimize the residual ‖AQ- B‖F over the Stiefel manifold of ort...We consider the unbalanced Procrustes problem with an orthonormal constraint on solutions： given matrices A ∈ R^n×n and B ∈ R^n×k, n 〉 k, minimize the residual ‖AQ- B‖F over the Stiefel manifold of orthonormal matrices. Theoretical analysis on necessary conditions and sufficient conditions for optimal solutions of the unbalanced Procrustes problem is given.展开更多
The necessary condition established in Part I of this paper for the global maximizers of the maximization problem max V tr（VTAV）/tr（VTBV）＋tr（VTCV）over the Stiefel manifold{V∈Rm×l ｜VTV=Il}（l〈m）,natural...The necessary condition established in Part I of this paper for the global maximizers of the maximization problem max V tr（VTAV）/tr（VTBV）＋tr（VTCV）over the Stiefel manifold{V∈Rm×l ｜VTV=Il}（l〈m）,naturally leads to a self-consistent-field（SCF）iteration for computing a maximizer.In this part,we analyze the global and local convergence of the SCF iteration,and show that the necessary condition for the global maximizers is fulfilled at any convergent point of the sequences of approximations generated by the SCF iteration.This is one of the advantages of the SCF iteration over optimization-based methods.Preliminary numerical tests are reported and show that the SCF iteration is very efficient by comparing with some manifold-based optimization methods.展开更多
We are concerned with the maximization of tr（V T AV）/tr（V T BV）＋tr（V T CV） over the Stiefel manifold {V ∈ R m×l ｜ V T V = Il} （l 〈 m）, where B is a given symmetric and positive definite matrix, A and...We are concerned with the maximization of tr（V T AV）/tr（V T BV）＋tr（V T CV） over the Stiefel manifold {V ∈ R m×l ｜ V T V = Il} （l 〈 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.展开更多
Let p be an odd prime. For the Stiefel manifold Wm＋k,k = SU（m ＋ k）/SU（m）, we obtain an upper bound of its p-primary homotopy exponent in the stable range k ≤ m with k ≤ （p - 1）2 ＋ 1.
We propose a new method for smoothly interpolating a given set of data points on Grassmann and Stiefel manifolds using a generalization of the De Casteljau algorithm.To that end,we reduce interpolation problem to the ...We propose a new method for smoothly interpolating a given set of data points on Grassmann and Stiefel manifolds using a generalization of the De Casteljau algorithm.To that end,we reduce interpolation problem to the classical Euclidean setting,allowing us to directly leverage the extensive toolbox of spline interpolation.The interpolated curve enjoy a number of nice properties:The solution exists and is optimal in many common situations.For applications,the structures with respect to chosen Riemannian metrics are detailed resulting in additional computational advantages.展开更多
基金the National Natural Science Foundation of China(Grant Nos.12125108,11971466,12288201,12021001,11991021)the Key Research Program of Frontier Sciences,Chinese Academy of Sciences(Grant No.ZDBS-LY-7022).
文摘In this paper,we present a novel penalty model called ExPen for optimization over the Stiefel manifold.Different from existing penalty functions for orthogonality constraints,ExPen adopts a smooth penalty function without using any first-order derivative of the objective function.We show that all the first-order stationary points of ExPen with a sufficiently large penalty parameter are either feasible,namely,are the first-order stationary points of the original optimization problem,or far from the Stiefel manifold.Besides,the original problem and ExPen share the same second-order stationary points.Remarkably,the exact gradient and Hessian of ExPen are easy to compute.As a consequence,abundant algorithm resources in unconstrained optimization can be applied straightforwardly to solve ExPen.
文摘We consider the unbalanced Procrustes problem with an orthonormal constraint on solutions： given matrices A ∈ R^n×n and B ∈ R^n×k, n 〉 k, minimize the residual ‖AQ- B‖F over the Stiefel manifold of orthonormal matrices. Theoretical analysis on necessary conditions and sufficient conditions for optimal solutions of the unbalanced Procrustes problem is given.
基金Acknowledgements The first author was supported by National Natural Science Foundation of China(Grant Nos.11101257 and 11371102)the Basic Academic Discipline Program,the 11th Five Year Plan of 211 Project for Shanghai University of Finance and Economics+1 种基金supported by National Science Foundation of USA(Grant Nos.1115834and 1317330)a Research Gift Grant from Intel Corporation
文摘The necessary condition established in Part I of this paper for the global maximizers of the maximization problem max V tr（VTAV）/tr（VTBV）＋tr（VTCV）over the Stiefel manifold{V∈Rm×l ｜VTV=Il}（l〈m）,naturally leads to a self-consistent-field（SCF）iteration for computing a maximizer.In this part,we analyze the global and local convergence of the SCF iteration,and show that the necessary condition for the global maximizers is fulfilled at any convergent point of the sequences of approximations generated by the SCF iteration.This is one of the advantages of the SCF iteration over optimization-based methods.Preliminary numerical tests are reported and show that the SCF iteration is very efficient by comparing with some manifold-based optimization methods.
基金supported by National Natural Science Foundation of China(Grant Nos.11101257 and 11371102)the Basic Academic Discipline Program+3 种基金the 11th Five Year Plan of 211 Project for Shanghai University of Finance and Economicsa visiting scholar at the Department of Mathematics,University of Texas at Arlington from February 2013 toJanuary 2014supported by National Science Foundation of USA(Grant Nos.1115834and 1317330)a Research Gift Grant from Intel Corporation
文摘We are concerned with the maximization of tr（V T AV）/tr（V T BV）＋tr（V T CV） over the Stiefel manifold {V ∈ R m×l ｜ V T V = Il} （l 〈 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.
基金Supported by the NSFC(Grant No.11261062)the Special Financial Grant form the China Postdoctoral Science Foundation(Grant No.2015T80909)Research Fund for the Doctoral Program of Higher Education of China(Grant No.20134407110001)
文摘Let p be an odd prime. For the Stiefel manifold Wm＋k,k = SU（m ＋ k）/SU（m）, we obtain an upper bound of its p-primary homotopy exponent in the stable range k ≤ m with k ≤ （p - 1）2 ＋ 1.
文摘We propose a new method for smoothly interpolating a given set of data points on Grassmann and Stiefel manifolds using a generalization of the De Casteljau algorithm.To that end,we reduce interpolation problem to the classical Euclidean setting,allowing us to directly leverage the extensive toolbox of spline interpolation.The interpolated curve enjoy a number of nice properties:The solution exists and is optimal in many common situations.For applications,the structures with respect to chosen Riemannian metrics are detailed resulting in additional computational advantages.
基金Supported by NSFC(11547122, 11571182, 11901300)NSFC Tianyuan Visiting Scholar Program in Mathematics(12226339)+1 种基金The Key Research Project of Henan Higher Education Institutions(China)(22A110021)Nanhu Scholars Program for Young Scholars of XYNU。
