Homotopy Exponents of Stiefel Manifolds in the Stable Range

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.

BEZIER SPLINES INTERPOLATION ON STIEFEL AND GRASSMANN MANIFOLDS

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.

SOLVING OPTIMIZATION PROBLEMS OVER THE STIEFEL MANIFOLD BY SMOOTH EXACT PENALTY FUNCTIONS

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.

Maximization of the sum of the trace ratio on the Stiefel manifold, II: Computation

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.

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

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.

CONDITIONS FOR OPTIMAL SOLUTIONS OF UNBALANCED PROCRUSTES PROBLEM ON STIEFEL MANIFOLD

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.

On Embeddings of Tori in Euclidean Spaces

Using the relation between the set of embeddings of tori into Euclideanspaces modulo ambient isotopies and the homotopy groups of Stiefel manifolds, we prove new resultson embeddings of tori into Euclidean spaces.

ACCELERATED OPTIMIZATION WITH ORTHOGONALITY CONSTRAINTS

We develop a generalization of Nesterov’s accelerated gradient descent method which is designed to deal with orthogonality constraints.To demonstrate the effectiveness of our method,we perform numerical experiments which demonstrate that the number of iterations scales with the square root of the condition number,and also compare with existing state-of-the-art quasi-Newton methods on the Stiefel manifold.Our experiments show that our method outperforms existing state-of-the-art quasi-Newton methods on some large,ill-conditioned problems.