Bin Gao

Bin Gao
University of Münster | WWU · Applied Mathematics: Institute for Analysis and Numerics

Postdoc

About

19
Publications
1,722
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113
Citations
Introduction
Please visit my personal homepage for more information: https://www.gaobin.cc/
Additional affiliations
September 2014 - July 2019
Chinese Academy of Sciences
Position
  • PhD

Publications

Publications (19)
Preprint
Full-text available
We study a type of Riemannian gradient descent (RGD) algorithm, designed through Riemannian preconditioning, for optimization on $\mathcal{M}_k^{m\times n}$ -- the set of $m\times n$ real matrices with a fixed rank $k$. Our analysis is based on a quotient geometric view of $\mathcal{M}_k^{m\times n}$: by identifying this set with the quotient manif...
Preprint
Full-text available
We study a continuous-time system that solves the optimization problem over the set of orthogonal matrices, which is also known as the Stiefel manifold. The resulting optimization flow follows a path that is not always on the manifold but eventually lands on the manifold. We introduce a generalized Stiefel manifold to which we extend the canonical...
Article
Full-text available
The low-rank matrix completion problem can be solved by Riemannian optimization on a fixed-rank manifold. However, a drawback of the known approaches is that the rank parameter has to be fixed a priori. In this paper, we consider the optimization problem on the set of bounded-rank matrices. We propose a Riemannian rank-adaptive method, which consis...
Chapter
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 Riemannia...
Preprint
Full-text available
The low-rank matrix completion problem can be solved by Riemannian optimization on a fixed-rank manifold. However, a drawback of the known approaches is that the rank parameter has to be fixed a priori. In this paper, we consider the optimization problem on the set of bounded-rank matrices. We propose a Riemannian rank-adaptive method, which consis...
Preprint
Full-text available
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...
Preprint
Full-text available
We propose new Riemannian preconditioned algorithms for low-rank tensor completion via the polyadic decomposition of a tensor. These algorithms exploit a non-Euclidean metric on the product space of the factor matrices of the low-rank tensor in the polyadic decomposition form. This new metric is designed using an approximation of the diagonal block...
Preprint
Full-text available
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 prob...
Article
The symplectic Stiefel manifold, denoted by Sp(2p; 2n), is the set of linear symplectic maps between the standard symplectic spaces R^{2p} and R^{2n}. When p = n, it reduces to the well-known set of 2nx2n symplectic matrices. Optimization problems on Sp(2p; 2n) find applications in various areas, such as optics, quantum physics, numerical linear al...
Preprint
Full-text available
We propose a class of multipliers correction methods to minimize a differentiable function over the Stiefel manifold. The proposed methods combine a function value reduction step with a proximal correction step. The former one searches along an arbitrary descent direction in the Euclidean space instead of a vector in the tangent space of the Stiefe...
Preprint
Full-text available
All-electron calculations play an important role in density functional theory, in which improving computational efficiency is one of the most needed and challenging tasks. In the model formulations, both nonlinear eigenvalue problem and total energy minimization problem pursue orthogonal solutions. Most existing algorithms for solving these two mod...
Preprint
Full-text available
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, su...
Article
Full-text available
To construct a parallel approach for solving optimization problems with orthogonality constraints is usually regarded as an extremely difficult mission, due to the low scalability of the orthonormalization procedure. However, such a demand is particularly huge in some application areas such as materials computation. In this paper, we propose a prox...
Preprint
Full-text available
To construct a parallel approach for solving optimization problems with orthogonality constraints is usually regarded as an extremely difficult mission, due to the low scalability of the orthonormalization procedure. However, such demand is particularly huge in some application areas such as materials computation. In this paper, we propose a proxim...
Article
Full-text available
In this paper, we consider a class of optimization problems with orthogonality constraints, the feasible region of which is called the Stiefel manifold. Our new framework combines a function value reduction step with a correction step. Different from the existing approaches, the function value reduction step of our algorithmic framework searches al...
Article
Full-text available
In this paper, we prove that the global version of the {\L}ojasiewicz gradient inequality holds for quadratic sphere constrained optimization problem with exponent $\theta=\frac{3}{4}$. An example from Ting Kei Pong shows that $\theta=\frac{3}{4}$ is tight. This is the first {\L}ojasiewicz gradient inequality established for the sphere constrained...

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