scholarly journals A Matrix-Algebraic Algorithm for the Riemannian Logarithm on the Stiefel Manifold under the Canonical Metric

2017 ◽  
Vol 38 (2) ◽  
pp. 322-342 ◽  
Author(s):  
Ralf Zimmermann
Keyword(s):  
Stiefel Manifold ◽  
Canonical Metric

scholarly journals Whitehead products in the complex Stiefel manifolds

1983 ◽  
Vol 26 (2) ◽  
pp. 241-251 ◽  
Author(s):  
Yasukuni Furukawa
Keyword(s):  
Stiefel Manifold ◽  
Isotropy Group ◽  
Homotopy Groups ◽  
Stiefel Manifolds

The complex Stiefel manifoldWn,k, wheren≦k≦1, is a space whose points arek-frames inCn. By using the formula of McCarty [4], we will make the calculations of the Whitehead products in the groups π*(Wn,k). The case of real and quaternionic will be treated by Nomura and Furukawa [7]. The product [[η],j1l] appears as generator of the isotropy group of the identity map of Stiefel manifolds. In this note we use freely the results of the 2-components of the homotopy groups of real and complex Stiefel manifolds such as Paechter [8], Hoo-Mahowald [1], Nomura [5], Sigrist [9] and Nomura-Furukawa [6].

scholarly journals Blind Data Detection in Massive MIMO via ℓ3-norm Maximization over the Stiefel Manifold

2020 ◽  
pp. 1-1
Author(s):  
Ye Xue ◽  
Yifei Shen ◽  
Vincent Lau ◽  
Jun Zhang ◽  
Khaled B. Letaief
Keyword(s):  
Massive Mimo ◽  
Stiefel Manifold ◽  
Data Detection

scholarly journals Clustering of cancer data based on Stiefel manifold for multiple views

2021 ◽  
Vol 22 (1) ◽  
Author(s):  
Jing Tian ◽  
Jianping Zhao ◽  
Chunhou Zheng
Keyword(s):  
Optimization Problem ◽  
Nearest Neighbor ◽  
Search Algorithm ◽  
Stiefel Manifold ◽  
Omics Data ◽  
K Nearest Neighbor ◽  
Cancer Data ◽  
Clustering Problem ◽  
Multiple Datasets ◽  
Cluster Class

Abstract Background In recent years, various sequencing techniques have been used to collect biomedical omics datasets. It is usually possible to obtain multiple types of omics data from a single patient sample. Clustering of omics data plays an indispensable role in biological and medical research, and it is helpful to reveal data structures from multiple collections. Nevertheless, clustering of omics data consists of many challenges. The primary challenges in omics data analysis come from high dimension of data and small size of sample. Therefore, it is difficult to find a suitable integration method for structural analysis of multiple datasets. Results In this paper, a multi-view clustering based on Stiefel manifold method (MCSM) is proposed. The MCSM method comprises three core steps. Firstly, we established a binary optimization model for the simultaneous clustering problem. Secondly, we solved the optimization problem by linear search algorithm based on Stiefel manifold. Finally, we integrated the clustering results obtained from three omics by using k-nearest neighbor method. We applied this approach to four cancer datasets on TCGA. The result shows that our method is superior to several state-of-art methods, which depends on the hypothesis that the underlying omics cluster class is the same. Conclusion Particularly, our approach has better performance than compared approaches when the underlying clusters are inconsistent. For patients with different subtypes, both consistent and differential clusters can be identified at the same time.

Cholesky QR-based retraction on the generalized Stiefel manifold

2018 ◽  
Vol 72 (2) ◽  
pp. 293-308 ◽  
Author(s):  
Hiroyuki Sato ◽  
Kensuke Aihara
Keyword(s):  
Stiefel Manifold

scholarly journals Statistics on the Stiefel manifold: Theory and applications

2019 ◽  
Vol 47 (1) ◽  
pp. 415-438 ◽  
Author(s):  
Rudrasis Chakraborty ◽  
Baba C. Vemuri
Keyword(s):  
Stiefel Manifold ◽  
Manifold Theory

Homogeneous Einstein Finsler Metrics on -dimensional Spheres

2018 ◽  
Vol 62 (3) ◽  
pp. 509-523
Author(s):  
Libing Huang ◽  
Xiaohuan Mo
Keyword(s):  
Sectional Curvature ◽  
Ricci Curvature ◽  
Einstein Metrics ◽  
Constant Sectional Curvature ◽  
Einstein Metric ◽  
Finsler Metrics ◽  
Dimensional Sphere ◽  
Order Ordinary Differential Equation ◽  
Canonical Metric ◽  
Homogeneous Einstein Metrics

AbstractIn this paper, we study a class of homogeneous Finsler metrics of vanishing $S$-curvature on a $(4n+3)$-dimensional sphere. We find a second order ordinary differential equation that characterizes Einstein metrics with constant Ricci curvature $1$ in this class. Using this equation we show that there are infinitely many homogeneous Einstein metrics on $S^{4n+3}$ of constant Ricci curvature $1$ and vanishing $S$-curvature. They contain the canonical metric on $S^{4n+3}$ of constant sectional curvature $1$ and the Einstein metric of non-constant sectional curvature given by Jensen in 1973.

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