A Grassmann manifold handbook: basic geometry and computational aspects
DOI10.1007/s10444-023-10090-8arXiv2011.13699OpenAlexW4390613691MaRDI QIDQ6184997
Ralf Zimmermann, Thomas Bendokat, Pierre-Antoine Absil
Publication date: 29 January 2024
Published in: Advances in Computational Mathematics (Search for Journal in Brave)
Full work available at URL: https://arxiv.org/abs/2011.13699
singular value decompositionorthogonal groupStiefel manifoldcurvaturecut locusgeodesicGrassmann manifoldparallel transportRiemannian exponentialconjugate locusquotient manifoldRiemannian logarithm
Grassmannians, Schubert varieties, flag manifolds (14M15) Eigenvalues, singular values, and eigenvectors (15A18) Research exposition (monographs, survey articles) pertaining to differential geometry (53-02) Affine differential geometry (53A15) Polar geometry, symplectic spaces, orthogonal spaces (51A50) Affine analytic geometry (51N10) Descriptive geometry (51N05) Research exposition (monographs, survey articles) pertaining to geometry (51-02) Orthogonal matrices (15B10)
Cites Work
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Algorithmic advances in Riemannian geometry and applications. For machine learning, computer vision, statistics, and optimization
- Optimization on a Grassmann manifold with application to system identification
- Geometric mean and geodesic regression on Grassmannians
- Introduction to the theory of Lie groups. Translated from the French by Urmie Ray
- On the largest principal angle between random subspaces
- Local convergence of an algorithm for subspace identification from partial data
- Model order reduction of parameterized circuit equations based on interpolation
- On cut loci of compact symmetric spaces
- On the geometry of complex Grassmann manifold, its noncompact dual and coherent states
- Riemannian geometry of Grassmann manifolds with a view on algorithmic computation
- Statistics on the Stiefel manifold: theory and applications
- Optimization and dynamical systems
- Data fitting on manifolds with composite Bézier-like curves and blended cubic splines
- Geodesic regression and the theory of least squares on Riemannian manifolds
- Low-rank matrix completion via preconditioned optimization on the Grassmann manifold
- Variations on a theme of Schubert calculus
- Zur Riemannschen Geometrie in Grassmannschen Mannigfaltigkeiten
- Why Use Sobolev Metrics on the Space of Curves
- Schubert Varieties and Distances between Subspaces of Different Dimensions
- On the Convergence of Gradient Descent for Finding the Riemannian Center of Mass
- A Locally Parametrized Reduced-Order Model for the Linear Frequency Domain Approach to Time-Accurate Computational Fluid Dynamics
- Introduction to Smooth Manifolds
- On evaluating higher-order derivatives of the QR decomposition of tall matrices with full column rank in forward and reverse mode algorithmic differentiation
- A real time procedure for affinely dependent parametric model order reduction using interpolation on Grassmann manifolds
- Geometric Methods and Applications
- Grassman manifolds as subsets of Euclidean spaces
- Grassmann manifolds and the Grassmann image of submanifolds
- Decomposition of Cut Loci
- The Geometry of Algorithms with Orthogonality Constraints
- On Smooth Decompositions of Matrices
- Introduction to Riemannian Manifolds
- Geometric Subspace Updates with Applications to Online Adaptive Nonlinear Model Reduction
- System- and Data-Driven Methods and Algorithms
- Hermite Interpolation and Data Processing Errors on Riemannian Matrix Manifolds
- Convergence Analysis of Gradient Algorithms on Riemannian Manifolds without Curvature Constraints and Application to Riemannian Mass
- Local improvements to reduced-order models using sensitivity analysis of the proper orthogonal decomposition
- Lie Groups and Geometric Aspects of Isometric Actions
- Lie Groups, Lie Algebras, and Representations
- Functions of Matrices
- Dynamical Low‐Rank Approximation
- Geometric Numerical Integration
- Unitarily Invariant Metrics on the Grassmann Space
- Multiscale Representations for Manifold-Valued Data
- DIFFERENTIAL GEOMETRY OF GRASSMANN MANIFOLDS
- Conjugate loci in Grassmann manifolds
- SECTIONAL CURVATURES OF GRASSMANN MANIFOLDS
- An accelerated first-order method for non-convex optimization on manifolds
