For questions about Stiefel-manifolds, the set of all orthonormal $k$-frames in $\Bbb R^n$.
Questions tagged [stiefel-manifolds]
78 questions
18
votes
1 answer
Fundamental groups of Grassmann and Stiefel manifolds
Could someone provide details on how to compute fundamental groups of real and complex Grassmann and Stiefel manifolds?
google
- 183
14
votes
1 answer
Difference between Grassmann and Stiefel manifolds
I'm working on an optimization problem on manifolds and I'm having a bit of a conceptual issue with choosing between the Grassmann and Stiefel manifolds. Grassmann(2, 3) is the linear subspace of dimension 2 within the space $\mathbb{R}^3$, so all…
bibliolytic
- 222
9
votes
1 answer
Is the Stiefel manifold $V_k(\mathbb{R}^n)$ homeomorphic to $O(n)/O(n-k)$?
Am I right that the Stiefel manifold $V_k(\mathbb{R}^n)$ (set of all orthonormal $k$-frames in $\mathbb{R}^n$) is homeomorphic to $O(n)/O(n-k)$?
Sketch of proof: any element of $V_k(\mathbb{R}^n)$ can be obtained from standard basis $\mathbb{R}^n$…
Aspirin
- 5,889
9
votes
1 answer
Obstruction cocycle of Stiefel manifold
I was reading about the interpretation of Stiefel Whitney classes as obstructions from Milnor-Stasheff's book and I got stuck at a step. The context is the following. Let $E \to B$ be a vector bundle of rank $n$ and let $V_k(\mathbb{R}^n)$ be the…
Manuel
- 1,778
7
votes
2 answers
Nearest semi-orthogonal matrix using the entry-wise $ {\ell}_{1} $ norm
Given an $m \times n$ matrix $M$ ($m \geq n$), the nearest semi-orthogonal matrix problem in $m \times n$ matrix $R$ is
$$\begin{array}{ll} \text{minimize} & \| M - R \|_F\\ \text{subject to} & R^T R = I_n\end{array}$$
A solution can be found by…
Francis
- 823
7
votes
1 answer
Geodesic on Stiefel manifold
Define a metric on Stiefel manifold $V_{n,p}$ as
$$\langle \Delta_1,\Delta_2 \rangle = \operatorname{tr} \left( \Delta_1^T\left(I-\frac{1}{2}YY^T\right)\Delta_2 \right)$$
for all $\Delta_1, \Delta_2 \in T_Y V_{n,p}$. How to calculate the geodesic…
gaoxinge
- 4,634
6
votes
1 answer
Orientability of Stiefel manifold $V_2(\mathbb R^4)$
What is an easy proof of orientability of Stiefel manifold $V_2(\mathbb{R}^4)$ (pairs of orthonormal vectors from $\mathbb{R}^4$ - subset of $\mathbb{R}^8$)? All proofs I found deal with Lie groups and other complicated for me stuff. I suppose that…
Richard Wagner
- 136
5
votes
1 answer
Cell structure on Stiefel manifolds
Denote by $V_{n,k}$ the set of $k$-frames in $\mathbb{R}^n$. This is a Stiefel manifold. I am trying to understand the following statement of the cellular structure (from Mosher & Tangora's book page 42)
By a normal cell of $V_{n,k}$ we mean a cell…
Juan S
- 10,526
5
votes
1 answer
Equation for geodesic in manifold of orthogonal matrices
From Geodesic on Stiefel manifold, a geodesic (under the canonical metric) on the manifold of orthogonal matrices can be expressed as
$$Y(t) = Q e^{Xt} I$$
for some matrices $Q$ and $X$.
Does it follow that the geodesic between two orthogonal…
opt_learn
- 53
5
votes
0 answers
Stiefel manifold is a manifold
We consider the real Stiefel manifold
$$V_k (\mathbb{R}^n) := \left\{ (v_1, v_2, \dots, v_k) \in S^{n-1} \times \cdots \times S^{n-1} \mid \langle v_i, v_j \rangle = \delta_{ij} \right\}$$
I want to know how to show that $V_k(\mathbb{R}^n)$ is a…
user267839
- 9,217
5
votes
1 answer
Fiber bundle on Stiefel manifold
Let $V_{n}(\mathbb{C}^{k})$ the Stiefel manifold of $n$-frame in $\mathbb{C}^{k}$. We can see $V_{n}(\mathbb{C}^{k})$ as a subset of $n$ copies of the cartesian product $S^{2k-1} \times \cdots \times S^{2K-1}$. So we have a bundle
$…
ArthurStuart
- 51
- 1
4
votes
1 answer
Least-squares problem with orthonormality constraints
Given $y_1,\dots, y_n \in {\Bbb R}$, $w \in {\Bbb R}^d$, and $x_1,\dots, x_n \in {\Bbb R}^D$, how do we solve the following optimization problem
\begin{align}
\min_A \,\, \sum_{i=1}^n \left( y_i - w^T A^T x_i \right)^2\\
\text{subject to} \qquad…
gmravi
- 366
4
votes
1 answer
Map between Stiefel manifold and the Grassmannian
I'm working on problem 2-7 in Lee's introduction to Riemannian Manifolds and am having trouble on part (b):
Let $V_k(\mathbb{R}^n)$ be the Stiefel manifold and $G_k(\mathbb{R}^n)$ denote the Grassmannian. I want to show that the map
$$ \pi :…
ABC
- 379
4
votes
5 answers
Maximize $\mathrm{tr}(Q^TCQ)$ subject to $Q^TQ=I$
Let $C \in \mathbb{R}^{d \times d}$ be symmetric, and
$$Q = \begin{bmatrix}
\vert & \vert & & \vert \\
q_1 & q_2 & \dots & q_K \\
\vert & \vert & & \vert
\end{bmatrix} \in \mathbb{R}^{d\times K}$$
where $d \geq…
abcd
- 299
4
votes
1 answer
Minimize $ \mbox{tr} ( X^T A X ) + \lambda \mbox{tr} ( X^T B ) $ subject to $ X^T X = I $ - Linear Matrix Function with Norm Equality Constraint
We have the following optimization problem in tall matrix $X \in\mathbb R^{n \times k}$
$$\begin{array}{ll} \text{minimize} & \mbox{tr}(X^T A X) + \lambda \,\mbox{tr}(X^T B)\\ \text{subject to} & X^T X = I_k\end{array}$$
where $A \in \mathbb R^{n…
E.J.
- 969