Numerous problems in optics, quantum physics, stability analysis, and control of dynamical systems can be brought to an optimization problem with matrix variable subjected to the symplecticity constraint. As this constraint nicely forms a so-called symplectic Stiefel manifold, Riemannian optimization is preferred, because one can borrow ideas from unconstrained optimization methods after preparing necessary geometric tools. Retraction is arguably the most important one which decides the way iterates are updated given a search direction. Two retractions have been constructed so far: one relies on the Cayley transform and the other is designed using quasi-geodesic curves. In this paper, we propose a new retraction which is based on an SR matrix decomposition. We prove that its domain contains the open unit ball which is essential in proving the global convergence of the associated gradient-based optimization algorithm. Moreover, we consider three applications--symplectic target matrix problem, symplectic eigenvalue computation, and symplectic model reduction of Hamiltonian systems--with various examples. The extensive numerical comparisons reveal the strengths of the proposed optimization algorithm.
Recursive Gradient Retraction Methods for Stochastic Nonconvex Optimization on Stiefel Manifolds
School of Systems Science and Engineering, Sun Yat-sen University（中山大学系统科学与工程学院）；Faculty of Computer Science and Artificial Intelligence, Shenzhen University of Advanced Technology（深圳先进技术大学计算机科学与人工智能学院）
Optimization without Retraction on the Random Generalized Stiefel Manifold
ICTEAM Institute, UCLouvain（ICTEAM学院，UCLouvain大学）；Department of Statistics, University of Oxford（统计系，牛津大学）；Apple Machine Learning Group（苹果机器学习小组）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（数学与系统科学学院，中国科学院）
The symplectic Stiefel manifold, denoted by $\mathrm{Sp}(2p,2n)$, is the set of linear symplectic maps between the standard symplectic spaces $\mathbb{R}^{2p}$ and $\mathbb{R}^{2n}$. When $p=n$, it reduces to the well-known set of $2n\times 2n$ symplectic matrices. We study the Riemannian geometry of this manifold viewed as a Riemannian submanifold of the Euclidean space $\mathbb{R}^{2n\times 2p}$. The corresponding normal space and projections onto the tangent and normal spaces are investigated. Moreover, we consider optimization problems on the symplectic Stiefel manifold. We obtain the expr…
The symplectic Stiefel manifold, denoted by $\mathrm{Sp}(2p,2n)$, is the set of linear symplectic maps between the standard symplectic spaces $\mathbb{R}^{2p}$ and $\mathbb{R}^{2n}$. When $p=n$, it reduces to the well-known set of $2n\times 2n$ symplectic matrices. Optimization problems on $\mathrm{Sp}(2p,2n)$ find applications in various areas, such as optics, quantum physics, numerical linear algebra and model order reduction of dynamical systems. The purpose of this paper is to propose and analyze gradient-descent methods on $\mathrm{Sp}(2p,2n)$, where the notion of gradient stems from a Ri…
Retractions maps are used to define a discretization of the tangent bundle of the configuration manifold as two copies of the configuration manifold where the dynamics take place. Such discretization maps can be conveniently lifted to the cotangent bundle so that symplectic integrators are constructed for Hamilton's equations. Optimal control problems are provided with a Hamiltonian framework by Pontryagin's Maximum Principle. That is why we use discretization maps and the integrability algorithm to obtain presymplectic integrators for optimal control problems.
Let $*$ denote the t-product between two third-order tensors. The purpose of this work is to study fundamental computation over the set $St(n,p,l):= \{\mathcal Q\in \mathbb R^{n\times p\times l} \mid \mathcal Q^{\top}* \mathcal Q = \mathcal I \}$, where $\mathcal Q$ is a third-order tensor of size $n\times p \times l$ and $\mathcal I$ ($n\geq p$) is the identity tensor. It is first verified that $St(n,p,l)$ endowed with the usual Frobenius norm forms a Riemannian manifold, which is termed as the (third-order) \emph{tensor Stiefel manifold} in this work. We then derive the tangent space, Rieman…
We consider the problem of minimizing a non-convex function over a smooth manifold $\mathcal{M}$. We propose a novel algorithm, the Orthogonal Directions Constrained Gradient Method (ODCGM) which only requires computing a projection onto a vector space. ODCGM is infeasible but the iterates are constantly pulled towards the manifold, ensuring the convergence of ODCGM towards $\mathcal{M}$. ODCGM is much simpler to implement than the classical methods which require the computation of a retraction. Moreover, we show that ODCGM exhibits the near-optimal oracle complexities $\mathcal{O}(1/\varepsil…
We address the problem of computing the smallest symplectic eigenvalues and the corresponding eigenvectors of symmetric positive-definite matrices in the sense of Williamson's theorem. It is formulated as minimizing a trace cost function over the symplectic Stiefel manifold. We first investigate various theoretical aspects of this optimization problem such as characterizing the sets of critical points, saddle points, and global minimizers as well as proving that non-global local minimizers do not exist. Based on our recent results on constructing Riemannian structures on the symplectic Stiefel…
Decentralized Riemannian Conjugate Gradient Method on the Stiefel Manifold
Zhejiang University（浙江大学）；SGIT AI Lab, State Grid Corporation of China（国网公司SGIT人工智能实验室）；CFAR and IHPC, Agency for Science, Technology and Research（新加坡科技研究局CFAR与IHPC）；SCSE, NTU（南洋理工大学计算机科学与工程学院）
In the present paper, in order to fnd a singularity of a vector field defined on Riemannian manifolds, we present a new globalization strategy of Newton method and establish its global convergence with superlinear rate. In particular, this globalization generalizes for a general retraction the existing damped Newton's method. The presented global convergence analysis does not require any hypotesesis on singularity of the vector field. We applied the proposed method to solve the truncated singular value problem on the product of two Stiefel manifolds, the dextrous hand grasping problem on the c…
Improved Penalty Function Approaches for Optimization Problems with General Orthogonality
Guangxi University（广西大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））
Primal-Dual Methods for Nonsmooth Nonconvex Optimization with Orthogonality Constraints
H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology（佐治亚理工学院H. Milton Stewart工业与系统工程系）；Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong（香港中文大学系统工程与工程管理系）
