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Questions tagged [stiefel-manifolds]

For questions about Stiefel-manifolds, the set of all orthonormal $k$-frames in $\Bbb R^n$.

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What is the action of $O(k)$ on $V_k(\mathbb R^n)$ making it a principal bundle?

Let $V_k(\mathbb R^n)$ be the Stiefel manifold of ordered $k$-tuples of vectors in $\mathbb R^n$. I have seen in many places that $V_k(\mathbb R^n)$ is an $O(k)$ principal bundle over the Grassmanian ...
Chris's user avatar
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50 views

differentiate a mtrix function with respect to a matrix

I want to differentiate $f_{\theta}(\nabla_{\theta}(X))$ with respect to $\theta$, where $\theta$ is orthogonal $n \times p$ matrix (lies on Stiefel manifold) and $\nabla_{\theta}(X)$ is the ...
Hadi.Tabe's user avatar
0 votes
1 answer
53 views

Modified Bessel Function Identity (change of variable integrating on the sphere)

Can someone explain why the following identity regarding the modified Bessel function of the first kind holds? $$\int_{\mathbb{S}^{p-1}} e^{\kappa \mu^T\mathbf{x}}d\mathbf{x} = B\left(\frac{p-1}{2}, \...
AAAA's user avatar
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3 votes
1 answer
69 views

Minimizing distance between two orthonormal k-frames in $R^{k+1}$

I want to minimize the distance between two orthonormal k-frames in $R^{k+1}$. The first k-frame is $\{e_1,...,e_k\}$ where $e_i = (0,...,0,1,0,,,0)$ denote i'th unit vectors in $R^{k+1}$. The second ...
tooth0pasty's user avatar
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52 views

Retraction on non-compact Stiefel manifold

I have been looking online for the retraction of the non-compact Stiefel manifold (where $k <m$): $R_*^{m \times k} = \{ M \in \mathbb{R}^{m \times k} : \operatorname{rk}(M)=k \}$ as seen here (...
Hugo B's user avatar
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0 answers
59 views

Expected value of the projective metric between random orthogonal Stiefel matrices in $\mathbb{R}^{N \times k}$ equals $1 - \frac{k}{N}$

This is a call for help to the random-matrix-theory savvy people. I've observed the below equality in experiments, and have been looking for a proof in the RMT literature but couldn't find one. I'd ...
fr_andres's user avatar
  • 139
1 vote
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24 views

Nontrivial Monodromy of the Universal Stiefel Bundle (and $O(n)$-equivariant vector fields on spheres)

Note: I'm not allowed to embed images into my posts yet, so I've linked my diagrams instead. Throughout, we will make use of the following result. Fact. For $H$ a Lie subgroup of $G$, there is a ...
Baylee Schutte's user avatar
1 vote
1 answer
44 views

$2$-norm of orthogonal transformation on subgaussian vector

Suppose we have a vector $e\in \mathbb{R}^n$, and a matrix $Q\in\mathbb{R}^{n\times p}$ such that $Q^{\top}Q=I_p$, i.e., the columns of $Q$ are orthornormal. Specifically, $e$ has i.i.d. subgaussian ...
TNLI's user avatar
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0 answers
131 views

Solution of a minimization problem under Stiefel manifold

I have the following optimization problem \begin{equation} \begin{aligned} \min_{X \in \Bbb R^{n \times c}} \quad & \operatorname{tr} \left( X^T A X \right) \\ \text { subject to } \quad & X^...
Alaeddine Zahir's user avatar
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0 answers
85 views

Moments of uniform distribution on Stiefel manifold

Suppose I have a $p \times n$ (with $p \geq n$) matrix $\bf U$ such that ${\bf U}' {\bf U} = {\bf I}_{n}$ and that $\bf U$ is uniformly distributed on the Stiefel manifold $V_{n,p}$. I would like to ...
Olivier Besson's user avatar
1 vote
1 answer
171 views

Brockett cost function on the Stiefel manifold

In section 4.8 of Optimization Algorithms on Matrix Manifolds (3rd version), Absil, Mahony & Sepulchre define the cost function $$ f : \mbox{St} (p,n) \to \mathbb{R}, \qquad X \mapsto \mbox{tr} \...
Fred f's user avatar
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1 answer
61 views

Can a tall matrix with orthonormal columns have rows of the same norm?

Let $n \geq r$ be two natural numbers and suppose $U \in \Bbb R^{n\times r}$ has orthonormal columns, i.e., $U^T U = I_r$. For any natural numbers $n \geq r$, is there a matrix $U \in \Bbb R^{n \times ...
Yunfei's user avatar
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3 votes
1 answer
105 views

Nearest semi-orthogonal matrix with fixed row

I want to find the nearest (in the Frobenius sense) $n\times d$ (where $n > d$) matrix $X$ to a given matrix $A$ of the same size such that the $d$ columns of $X$ are orthonormal . Since $X$ would ...
Arman's user avatar
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0 answers
135 views

Solving a variant of an orthogonal Procrustes problem

While reading the following paper on an optimization problem, there was a variant of an orthogonal Procrustes problem, where the solution is an element of the Stiefel manifold. The authors provided a ...
schlodinger's user avatar
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1 answer
62 views

Is $\{X\in\mathbb{R}^{n\times p}|X^TX=I, XX^T\circ I=\dfrac{p}{n}I\}$ a submanifold of $\mathbb{R}^{n\times p}$?

$\{X\in\mathbb{R}^{n\times p}|X^TX=I\}$ is a Stiefel manifold and this is known as a submanifold of $\mathbb{R}^{n\times p}$. However, if we add the additional condition $XX^T\circ I=\dfrac{p}{n}I$, ...
Jeong's user avatar
  • 1,458
1 vote
2 answers
984 views

Projection onto the Stiefel manifold and the orthogonal Procrustes problem

Let $m$ and $n$ be positive integers such that $m \ge n$, the case with $m >n$ being particularly interesting to us. The Stiefel manifold is \begin{equation} \mathbb{S}^{m, n} = \{X \in \mathbb{R}^{...
Nuno's user avatar
  • 668
2 votes
0 answers
108 views

Volume of a rescaled Stiefel manifold

The volume of the complex Stiefel manifold (where $n>p$) $$\mathcal{S}(n,p)=\{\boldsymbol{Q}\in\mathbb{C}^{n\times p}|\boldsymbol{Q}^{\rm H}\boldsymbol{Q}=\boldsymbol{I}\} $$ is given by $$ {\rm ...
Tucker Yuan's user avatar
0 votes
1 answer
131 views

Symmetry property of the Cayley transform on Stiefel manifolds

Context I am interested in the Cayley transform on the Stiefel manifold $\mathcal V_{np}$ at the point $I_{np} = (I_p, 0_{p,n-p})^\top$. For a matrix $X$ of the form $$ X = \begin{pmatrix} A & -B^\...
laotseu's user avatar
  • 43
4 votes
1 answer
513 views

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 ...
ABC's user avatar
  • 379
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0 answers
88 views

Stiefel manifolds & number of free parameters

Recently, we got introduced to the Stiefel manifold $V_k(\mathbb R^n)$, that is, the set of orthonormal $k$-frames in $\mathbb R^n$. In particular, we discussed the case for $k=2$, $V_2(\mathbb R^n) \...
Zest's user avatar
  • 2,448
2 votes
1 answer
151 views

Optimizing the sum of quadratic forms linked by orthogonality constraints

I am looking to find a qualitative solution to the optimization problem: $$\text{min}_{\{\mathbf{u}_i\}_i}\quad\sum_i \mathbf{u}_i^T\mathbf{M}_i\mathbf{u}_i \\ \text{s.t.}\quad \mathbf{u}_i^T\mathbf{u}...
mrs's user avatar
  • 29
0 votes
0 answers
321 views

What is the dimension of a complex Stiefel manifold

The Stiefel manifold is defined as follows. \begin{equation} V_{k}(\mathbb{C}^{n}) = \big\{ X \in \mathbb{C}^{n\times k} | X^{*} X = I_{k} \big\} \end{equation} where $n \geqslant k \geqslant 1$. ...
Liang Liao's user avatar
0 votes
1 answer
439 views

Tangent and normal space of the Stiefel manifold

Let $V_{n,p}$ be the real Stiefel manifold with parameters $n$ and $p$. Let $X \in V_{n,p}$ and $Y(t)$ curve on $V_{n,p}$ with $Y(0) = X$. Then, $\dot{Y}(0)$ is a tangent vector to $V_{n,p}$ at $X$. ...
Liang Liao's user avatar
1 vote
1 answer
260 views

Parallel transport on the Stiefel manifold

I am reading the paper Alan Edelman, Tomas A. Arias, Steven T. Smith, The geometry of algorithms with orthogonality constraints, SIAM Journal on Matrix Analysis and Applications, Volume 20, Number 2, ...
gcc's user avatar
  • 115
1 vote
0 answers
124 views

Haar measure from the Stiefel Manifold

I am reading paper Finite free convolutions of polynomials, which uses Haar meausre from the Stiefel manifold. All I know about measure theory is the Lebesgue measure on the Euclidean space. I want ...
Nate's user avatar
  • 715
1 vote
0 answers
222 views

collection of $m-$frame is a Manifold (Stiefel manifold)

Let $X = \mathbb{R}^{m+n}$ be the space. Define $$F_m(X) = \{(v_1, v_2, ..., v_m) : v_i \in X, \{v_1, v_2, ..., v_m\} \ \mbox{is linearly independent}\} \subseteq \mathbb{R}^{m+n} \times ... \times \...
user117375's user avatar
  • 1,211
3 votes
5 answers
1k views

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 &...
abcd's user avatar
  • 279
0 votes
1 answer
374 views

Stiefel manifold is homeomorphic to $O(n)/O(n - k)$

I have a question on Is the Stiefel manifold $V_k(\mathbb{R}^n)$ homeomorphic to $O(n)/O(n-k)$? Why is the described map bijective? The statement '$A \sim B \iff AB^T \in O(n - k)$ so $V_k(\mathbb{R}^...
user388557's user avatar
  • 2,554
2 votes
1 answer
179 views

Optimizing Trace$(Q^TZ)$ subject to $Q^TQ=I$

Let $Z \in \mathbb{R}^{m \times n}$ be a tall matrix ($m > n$). Solve the following optimization problem in $Q \in \mathbb{R}^{m \times n}$ $$\begin{array}{ll} \text{maximize} & \mbox{Tr} \left(...
kkcocoqq's user avatar
  • 306
1 vote
0 answers
192 views

Grassmann manifold

Grassmann manifold $G_{k,n}$ is the set of k-dimensional subspaces of $\mathbb{R^n}$. Let’s consider the set of k-frames $V_{k,n}$. I want to show that $$ G \to V_{k,n} \to^{\pi} G_{k,n} $$ can be ...
Klara Yitsovich's user avatar
1 vote
1 answer
429 views

Trace minimization in a Rayleigh-quotient-like problem

Given an $n\times n$ real diagonal matrix $D$ and an $m\times m$ real diagonal matrix $W$ (where $n\geq m$) with $\text{tr}(W^2)=1$, consider the following optimization problem in $X \in \mathbb{R}^{n ...
sunga's user avatar
  • 11
2 votes
3 answers
278 views

How to solve the system of matrix equations $XX^TA = A$, $X^TX = I$?

Given tall matrix $A \in \mathbb R^{n \times k}$ (where $n \gg k$), is there a way to solve the following system of matrix equations in $X \in \mathbb R^{n \times k}$? $$\begin{aligned} X X^T A &=...
Youwei Liang's user avatar
2 votes
1 answer
465 views

Maximize trace over Stiefel manifold

This question is the same as the question in this post. The OP of that post changed what they were asking and reduced it to a special case, so I’m asking the question in full generality here. Given ...
David M.'s user avatar
  • 2,633
1 vote
1 answer
34 views

References for injectivity/surjectivity $\pi_i(U(n-k)) \rightarrow \pi_i(U(n))$

I have read (Theorem 29.3, line 7, pg 144) the following statement: $\pi_i(U(n-k)) \rightarrow \pi_i(U(n))$ is surjective for $i \le 2(n-k)+1$ and injective for $i \le 2(n-k)$. and ...
Bryan Shih's user avatar
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2 votes
1 answer
52 views

Smallest trace of a matrix product where one is given and the other is orthogonal

What are the optimal solution and optimal value for the following semidefinite program $$ \min_{ V } \{ \mbox{tr} (V\Sigma) : VV^T=I \}$$ where $\Sigma$ is a given positive semidefinite matrix, and ...
O. Richard's user avatar
3 votes
1 answer
1k views

Uniform distribution on Stiefel manifold

I want to implement the method of sampling (uniformly) points on Stiefel manifold but I'm failing to find any kind of research/article/work that can give some info about the methods and techniques of ...
user2660964's user avatar
4 votes
0 answers
417 views

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 ...
user267839's user avatar
  • 7,499
3 votes
1 answer
229 views

How to optimize objective in the Grassmann manifold?

For Stiefel manifold, it contains all the orthogonal column matrices $$St(d,M) = \{X \in R^{M \times d} | X^TX = I\}$$ For Grassmann manifold, it is $$Gr(d,M) = \{col(X), X \in R^{M \times d}\}$$ ...
jason's user avatar
  • 837
1 vote
0 answers
254 views

'Jacobian' of QR decomposition of a rectangular matrix

I want to calculate the volume of real Stiefel manifold $V_{k}(\mathbb{R}^N)$ . $$ V_{k} (\mathbb{R}^N) = \{ H \in M(N, k, \mathbb{R})| H^{T}H = I_{k} \} $$ ((^T) denotes transposed matrix. $M(N, k, \...
T. Phan's user avatar
  • 11
1 vote
0 answers
89 views

General Stiefel-Whitney classes and Stiefel manifolds

Here are some statements that I wish to understand more deeply, whose truth value I want to check, and to determine under which criteria they are valid. Consider the Stiefel manifold $V_k(R^n)$ of ...
wonderich's user avatar
  • 5,969
3 votes
1 answer
181 views

Spheres and orthogonal matrices as spaces of solutions to matrix equations

For $a \in \mathbb{R}^n$, the solutions to $$1=\sum_{i=1}^na_i^2$$ form an $(n-1)$-sphere in $\mathbb{R}^n$. Meanwhile, for $A \in \mbox{GL}_k (\mathbb{R})$, the solutions to $$1=AA^T$$ are the ...
aaa's user avatar
  • 421
2 votes
1 answer
261 views

BO(-) example in Weiss Calculus

I'nm reading Orthogonal Calculus by Michael Weiss, and trying to understand example 2.7, concerning the derivatives of the functor $BO$, which sends a (finite dimensional) inner product space to the ...
Niall Taggart's user avatar
4 votes
1 answer
775 views

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} &...
E.J.'s user avatar
  • 949
1 vote
0 answers
61 views

From what manifold do we need to start to build an eversion for $any$-tridimensional object?

If a sphere eversion is possible using a half-way model how model is used for $cylinder$ eversion ? I need to make some premises to be able to frame the true nature of the problem In sphere eversion ...
user332153's user avatar
1 vote
0 answers
67 views

Self maps on Stiefel Manifolds

Let $m_l: S^{n-1} \stackrel{z \mapsto z^l}{\to} S^{n-1}$ be a map of spheres. Then it induces a map on $( S^{n-1})^k$, given by product. Note the Stiefel manifold $V_k(\mathbb{R^{n}})$, the ...
Surojit's user avatar
  • 881
2 votes
0 answers
82 views

Neighbourhood of a point in Stiefel manifold

In Hatcher above, he claims that "This determines orthonormal bases for the $(k-m)$ planes orthogonal to all nearby $m$-frames", I feel confused about this claim (even after checking the preceding ...
Danny's user avatar
  • 1,897
7 votes
2 answers
442 views

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} &...
Francis's user avatar
  • 823
4 votes
2 answers
714 views

Cohomology of Stiefel manifolds

Define the complex Stiefel space $W_{n,k}$ as $U(n)/U(k)$. What is its (co)homology? (Either singular or de Rham). I've searched through a bunch of classical references but can't seem to find this ...
Tim's user avatar
  • 3,419
13 votes
1 answer
3k views

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 ...
bibliolytic's user avatar
2 votes
0 answers
57 views

Show that Gram-Schmidt$(Y)$ = $B$Gram-Schmidt$(X)$ such that $B$ is an orthogonal matrix

Let $\begin{align} V_{k,n} &= \Bigl \{ \begin{pmatrix} v_{1} \\ v_{2} \\ \vdots \\ v_{k} \end{pmatrix} : v_i\in \mathbb{R}^n\text{ and } \{v_1,\...
sm10's user avatar
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